The Recursion Method (Lecture Notes in Physics monographs)

Lecture Notes in Physics New Series m: Monographs Editorial Board H.Araki Research Institute for Mathematical Sciences Kyoto University, Kitashirakawa Sakyo-ku, Kyoto 606, Japan E. Brezin Ecole Normale Superieure, Departement de Physique 24, rue Lhomond, F-75231 Paris Cedex 05, France J. Ehlers Max-Planck-Institut fUr Physik und Astrophysik, Institut fUr Astrophysik Karl-Schwarzschild-Strasse 1, D-85748 Garching, FRG U. Frisch Observatoire de Nice B. P. 229, F-06304 Nice Cedex 4, France

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v. S. Viswanath

Gerhard Muller

The Recursion Method Application to Many-Body Dynamics

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Authors V. S Viswanath Gerhard Muller Department of Physics The University of Rhode Island Kingston, RI 02881-0817, USA Email: [email protected]

ISBN 3-540-58319-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-58319-X Springer-Verlag New York Berlin Heidelberg CIP data applied for. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer -Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heide1berg 1994 Printed in Germany

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PREFACE AND ACKNOWLEDGMENTS This monograph is the product of our exploration of the recursion method as a general calculational technique for the study of quantum and classical many-body dynamics - a method of considerable versatility with an enormous potential for refinement and further development. What had accumulated in our minds and on our desks as a collection of facts about the recursion method and as an assortment of new results for a variety of applications, was not originally intended to be published in any form other than that of a series of research papers. However, our work on projects involving the recursion method took on a new dimension when one of us (GM) was invited to present a lecture series on this subject in Switzerland and in Germany during the fall of 1991. The first lecture series was given at the Ecole Polytechnique Federale in Lausanne, as part of the Troisieme Cycle de Physique des Universites Romandes. The second lecture series, given at the Institut fUr Physik der Universitat Dortmund, was sponsored by the Graduiertenkolleg Festkorperspektroskopie. Support from both institutions is gratefully acknowledged. The book may be divided into two parts of roughly equal size. The first part (Chapters 1 to 9) describes the different representations and formulations of the recursion method and its accessories, including various techniques of continuedfraction analysis, and recursive algorithms for the computation of ground-state wave functions. It contains numerous illustrations and simple applications. The second part (Chapters 10 and 11) is intended to illustrate the usefulness and importance of the recursion method in relation to other analytical and computational techniques as applied to two areas of current research in many-body dynamics. It reports recent results which we have obtained in a series of collaborative projects with several colleagues and students. We owe many thanks to H. Beck of the University of Neuchatel for suggesting and organizing the first lecture series, and to his collaborators at the time, M. B. Cibils (now in Lausanne) and Y. Cuche (now in Florence), for their kind assistance in many respects. No fewer thanks go to H. Keiter, U. Brandt, and W. Weber, for their kind hospitality at the University of Dortmund and, especially, to J. Stolze (now in Bayreuth), for organizing the second lecture series. Substantial parts of the manuscript were outlined and drafted at the Institut fUr Theoretische Physik at the University of Basel, where GM spent the larger part of his sabbatical leave from URI in the stimulating and pleasant environment around H. Thomas and his collaborators. Thank you very much for the kind hospitality! The research done in Basel was supported in part by the Swiss National Science Foundation and the research done at URI by the US National Science Foundation. Most of the computations were carried out on supercomputers at the National Center for Supercomputing Applications, University of TIlinois at UrbanaChampaign. Very special thanks go to M. H. Lee at the University of Georgia for having kindled our curiosity about the recursion method in the first place, and for his critical reading of the manuscript and his invaluable suggestions for improvements. We are much indebted to M. P. Nightingale, J.-M. Liu, S. Zhang, and Y. Yu of

VI

URI, to H. Leschke of the University of Erlangen-Niimberg and to N. Srivastava of Thinking Machines Corporation, for their contribution to the recent work portrayed in this book, and/or for their feedback from reading parts of the manuscript. Our warmest thanks go to Joachim Stolze of the University of Bayreuth, who came to URI for one year in the fall of 1992, and also to Markus Bohm of the University of Erlangen-Niimberg, who stayed at URI for one month in the spring of 1993. We have enjoyed very rewarding collaborations with these two colleagues and friends on several projects involving the recursion method. These collaborations have produced results of crucial importance for the shaping of the second part of the monograph. It would not have been possible to keep this project on track without the institutional support secured by S. S. Malik, Chairman of the Physics Department at URI, during a turbulent three years. The technical assistance provided by our systems manager, S. Pellegrino, has greatly facilitated the writing of this book and the computational research reported therein. Finally, we would like to express our thanks to the editor, Prof. W. BeiglbOck, for his most valuable advice, and to his coworkers at Springer-Verlag, Ms. S. Landgraf and Ms. B. Reichel-Mayer, for their expert help in the production of the camera-ready manuscript. Kingston, June 1994

V.S. Viswanath Gerhard Muller

CONTENTS INTRODUCTION 1-1 Calculational Techniques in Condensed Matter Theory 1-2 Recursion Method Applied to Many-Body Dynamics 1-3 Fonnalism and Goals 1-4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 6 9

2 LINEAR RESPONSE AND EQUILffiRIUM DYNAMICS . . . . . . . . . .. 11 2-1 Response Function and Generalized Susceptibility 11 2-2 Fluctuation-Dissipation Theorem 13 2-3 Moment Expansion 15 3 LIOUVILLIAN REPRESENTATION 3-1 Quantum Fonnulation . . . . . . . . . .. 3-2 Classical Fonnulation 3-3 Orthogonal Expansion of Dynamical Variables. . . . . . . . . . . .. 3-4 Relaxation Function and Spectral Density 3-5 Recursion Method and Moment Expansion 3-6 Generalized Langevin Equation 3-7 Projection Operator Fonnalism . . . . . . . . . . . . . . . . . . . . . . .. 3-8 Retarded Green's Functions

17 17 18 19 21 22 24 26 29

4 HAMILTONIAN REPRESENTATION 4-1 Orthogonal Expansion of Wave Functions 4-2 Structure Function 4-3 Continued-Fraction Coefficients and Frequency Moments 4-4 Lanczos Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 Modified Lanczos Method . . . . . . . . . . . . . . . . . . . . . . 4-6 Conjugate-Gradient Method . . . . . . . . . . . . . . . . . . . . . 4-7 Steepest-Desceht Method . . . . . . . . . . . . . . . . . . . . . . . 4-8 Comparative Perfonnance Test . . . . . . . . . . . . . . . . . . . 4-9 Green's Functions: Spectral and Continued-Fraction Representations

32 32 34 35 37 40 42 43 45

5 GENETIC CODE OF SPECTRAL DENSITIES 5-1 Finite ~k-Sequences 5-2 Spectral Densities with Bounded Support 5-3 Spectral Densities with Bounded Support and a Gap 5-4 Spectral Densities with Unbounded Support 5-5 Spectral Densities with Unbounded Support and a Gap 5-6 Orthogonal Polynomials 6 RECURSION METHOD ILLUSTRATED 6-1 Hannonic Oscillator 6-2 Spin Waves

. . . . .

. . . . .

. . . . .

.. .. .. .. ..

47 51 51 52 56 , 57 60 62 64 64 68

VIII

6-3 Lattice Fennions 69 6-4 Quantum Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73 6-5 Classical Spins 74 7 UNIVERSALITY CLASSES OF DYNAMICAL BEHAVIOR 7-1 Dynamics of the Equivalent-Neighbor XYZ Model 7-2 Fluctuation Functions and Spectral Densities for the XXZ Case 7-3 Recursion Method Applied to Equivalent-Neighbor Spin Models 7-4 Quantum Equivalent-Neighbor XYZ Model 7-5 Prototype Universality Classes 7-6 Two-Sublattice Spin Model with Long-Range Interaction 7-7 Many-Body Systems with Short-Range Interaction

76 76 78 80 82 82 87 90

8 TERMINATION OF CONTINUED FRACTIONS: ATTEMPTS AT DAMAGE CONTROL 93 8-1 Cut-Off Tennination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94 8-2 n-Pole Approximation 97 8-3 Pole Locations and Spectral Densities 98 8-4 Memory Functions and Fluctuation Functions 102 8-5 Moving Beyond Truncation 104 9 RECONSTRUCTION OF SPECTRAL DENSITIES FROM INCOMPLETE CONTINUED FRACTIONS 9-1 Model Tenninators from Model Spectral Densities 9-2 Square-Root Tenninator . . . . . . . . . . . . . . . . . . . . . . . . . .. 9-3 Rectangular Tenninator 9-4 Endpoint Singularities 9-5 ~-Tenninator................................... 9-6 Gap Tenninators 9-7 Infrared Singularities in Spectral Densities with Bounded Support 9-8 Spectral Densities with a &-Function Central Peak. . . . . . . .. 9-9 Tenninator with Matching Infrared Singularity 9-10 Compact a-Tenninator 9-11 Gaussian Tenninator . . .. 9-12 Infrared Singularities in Spectral Densities with Unbounded Support 9-13 Unbounded a-Terminator. . . . . . . . . . . . . . . . . . . . . . . . .. 9-14 Split-Gaussian Terminator 10

109 109 112 114 116 118 118 122 124 127 129 131 133 133 134

TRANSPORT OF SPIN FLUCTUATIONS AT HIGH TEMPERATURE 136 10-1 Generic High-Temperature Spin Dynamics 136 10-2 10 s=1/2 XYZ Model on Semi-Infinite Chain . . . . . . . . . .. 138

IX

10-3 Spin-1I2 XX Model: Neither Spin Diffusion nor Exponential Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10-4 Boundary Effects: Buildup of an Infrared Divergence 10-5 Boundary Effects: Crossover Between Growth Rates A.=O and A.=l 10-6 Spin-1I2 XXZ Model . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10-7 From Gaussian Decay to Exponential Decay , 10-8 Analysis of L\k-Sequences with Growth Rates near A.=1 10-9 From Exponential Relaxation to Diffusive Long-Time Tails 10-10 Sustained Power-Law Decay. . . . . . . . . . . . . . . . . . . . . .. 10-11 From Ballistic to Diffusive Transport of Spin Fluctuations .. 10-12 Boundary-Spin Spectral Densities 10-13 Spectral Signature of Quantum Spin Diffusion in Dimensions d=I,2,3 10-14 Spin Diffusion in the Classical Heisenberg Model. . . . . . .. 10-15 Is Classical Spin Diffusion Anomalous? . . . . . . . . . . . . . .. 10-16 Exponential DecayVersus Long-Time Tails 10-17 Anomalous Exponent or Non-Asymptotic Effect? 10-18 Experimental Evidence for Anomalous Spin Diffusion . . . .. 10-19 Two Kinds of Computational Errors 10-20 q-Dependent Correlation Function 10-21 Power Law Long-Time Tail with Logarithmic Correction .. 10-22 Effective Exponent 10-23 Effect of Exchange Inhomogeneities 11 QUANTUM SPIN DYNAMICS AT ZERO TEMPERATURE 11-1 ID s=1I2 XY Model with Magnetic Field. . . . . . . . . . . . .. 11-2 Product Ground State of the ID Spin-s XYZ Model with Magnetic Field 11-3 Conditions for the Existence of Linear Spin Waves 11-4 Resonances with Intrinsic Width 11-5 Finite and Infinite Bandwidths. . . . . . . . . . . . . . . . . . . . .. 11-6 Limitations of Single-Mode Picture 11-7 Spin Dynamics at Tc=O Critical Point: Exact Results for the Transverse Ising Model and the XX Model 11-8 Long-Time Asymptotic Expansions 11-9 Structure Functions and Their Singularities . . . . . . . . . . . .. 11-10 Structure Functions Reconstructed by Continued-Fraction Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11-11 Dynamic Structure Factors Szz(q,ro)n and Szz(q,ro)xx: Two-Particle Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .. 11-12 Dynamic Structure Factor S;xx(q,ro)xx: Continued-Fraction Analysis . . . . . . . . . . . . . . . . . . . . . .. 11-13 ID s=1I2 XYZ Model: Ground State and Excitation Spectrum

138 140 142 145 147 147 149 151 154 155 158 164 164 167 167 168 171 173 175 176 178 183 183 186 187 188 189 191 193 194 196 199 202 204 206

x 11-14 ID s= 112 XXZ Model: Criticality and Long-Range Order , 11-15 Excitation Spectrum of the ID s=1I2 XXZ Ferromagnet: Spin Waves and Bound States 11-16 Excitation Spectrum of the ID s=1I2 XXZ Antiferromagnet: Spinons and Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11-17 Equal-Time Correlation Functions 11-18 Dynamic Correlation Functions . . . . . . . . . . . . . . . . . . . .. 11-19 Continuum Approximation (Luttinger Model) 11-20 Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . .. 11-21 Weak-Coupling and Strong-Coupling Regimes 11-22 Infrared Singularities in SIJI.I(1t,ro) and ~Il(ro) . . . . . . . . . . .. 11-23 Reconstruction of Szz(1t,ro) (Weak-Coupling Analysis) 11-24 Reconstruction of Sxx(q,ro) (Strong-Coupling Analysis) 11-25 Strong-Coupling Reconstruction of SU(ro) 11-26 2D s= 1/2 XXZ Antiferromagnet 11-27 Dynamic Structure Factors Sxx(1t,1t,ro) and Szz(1t,1t,ro) . . . . .. 11-28 ID Spin-1 Heisenberg Antiferromagnet with Uniaxial Exchange and Single-Site Anisotropy " 11-29 Dynamically Relevant Excitation Gaps 11-30 Dispersion and Line Shapes BffiLIOGRAPHY

211 212 214 217 218 219 222 225 227 230 234 237 239 241 244 245 248 251

1 INTRODUCTION I-I Calculational Techniques in Condensed Matter Theory The toolbox of an experienced condensed matter theorist is divided into two major compartments, each one crowded with calculational techniques. One of the two compartments bears the label universal tools, the other one precision instruments. The universal-tools compartment contains an assortment of general methods for the calculation of observable quantities of interest in condensed matter physics. Among them are general methods for a particular purpose. General methods for the calculation of dynamic correlation functions which are applicable to arbitrarily selected model systems, for example, belong in that compartment. Also found there are multi-purpose methods with a wide range of applicability. Universal tools mus~ have a certain robustness against conditions that may invalidate their applicability. However, they are not meant to yield exact results on such a wide territory of applications. The general calculational or computational techniques that are commonly used in condensed matter theory may be categorized as follows: o Methods with extrinsic limitations, such as computer simulations, Green's function methods, the recursion method, or finite-size studies. In these methods, the limitations are set by the amount of calculational effort or computational power invested in them. o Methods with intrinsic limitations, such as mean field theory, linear spinwave theory (harmonic approximation), the random-phase approximation (within the framework of Green's function methods), the n-pole approximation (within the framework of the recursion method), have built-in limitations that cannot be overcome within their own respective scope. The precision instruments, stored in the other compartment of the theorist's toolbox, are a collection of special methods. They have been designed for the exact solution of particular problems. The small but precious collection of exactly solved models in statistical mechanics and solid state physics is their main source of origin. Once a challenging problem has been solved by a special method, it is by no means guaranteed that the solution can be reproduced by a general method. Nevertheless, it is usually instructive and illuminating to test the performance of universal tools on problems that have previously been solved by specially designed precision instruments. Many a general method has its root in a special method designed for the exact solution of a specific problem. What makes it a general method are applications to problems of a similar nature, where it is subject to intrinsic or extrinsic limitations. For example, mean field theory may be regarded as a special method for the solution of certain model systems with long-range interaction, and spin-wave theory derives its legitimacy from those special situations in which anharmonicities can be ignored in magnetic excitations. For a thorough study of a broadly defined topic we need universal tools and precision instruments, i.e. general methods with their wide range of applicability and special methods that provide deeper insight for particular circumstances. We must combine systematics in breadth with systematics in depth in order to gain the

2

Chapter 1

best possible understanding of the topic under scrutiny. The use in isolation of (i) general methods with severe intrinsic limitations or (ii) special methods applicable only under highly non-generic circumstances is likely to invite misleading conclusions. Systematics in both directions is key to an understanding of the least accessible territory. Exact solutions are usually out of reach except for particular circumstances and by special methods. The particular circumstances are always describable in terms of a simplification of the problem. There are basically two types of simplifications that may bring an exact solution to within reach: o Simplifications due to a special type of interaction between the degrees of freedom. The free-particle limit or special types of infinite-range interaction are obvious examples. o Simplifications due to a special state of a model system which otherwise exhibits generic behavior. A typical example is the ordered ground state of the Heisenberg ferromagnet. Do these particular, simplifying circumstances translate into an improved performance of applicable general methods as well? The answer depends on the specifics of the general method under consideration. (i) Green's function methods pose the problem of approximating the infinite hierarchy of equations of motion in a controlled and systematic way. That is notoriously difficult even for weakly coupled degrees of freedom. However, in the noninteracting limit, that hierarchy reduces to a closed set of equations, from which the exact solution can readily be extracted. Simplifications due to a special state of the system do not, in general, result in a more tractable hierarchy of equations of motion. The reason is that the state of the system remains unspecified in the hierarchy of equations of motion. (ii) In the recursion method, the properties of generic systems manifest themselves in highly complex patterns exhibited by the sequences of continued-fraction coefficients, as we shall see. For noninteracting degrees of freedom, the amount of simplification in those sequences is comparable to that in the Green's function approach. However, the recursion method is decidedly better equipped to handle situations in which the simplification is due to a special state of the system. The reason is that the specification of the state has its impact on every continuedfraction coefficient as it is evaluated in the recursive calculational procedure.

1-2 Recursion Method Applied to Many-Body Dynamics The link between almost any kind of experiment that probes dynamical properties of condensed matter systems and the calculational techniques used for studying equilibrium dynamics of quantum or classical many-body systems is the linear response theory (see Fig. 1-1). Its core consists of (i) the fluctuation-dissipation theorem in the most general form and (ii) the implications of the causality principle including the Kramers-Kronig dispersion relations.

Section 1-2

3

RECURSION METHOD

Figure 1-1: Linear response theory is the indispensable link between dynamics experiments

in condensed matter physics and the recursion method for applications to many-body dynamics or any other calculational technique for the study of systems in thermal equilibrium.

4

Chapter 1

The recursion method is one of several general methods for the calculation of dynamic correlation functions for model systems in equilibrium states. 1 The purpose of this book is to give a detailed description of the recursion method in its diverse representations and formulations (see Fig. 1-1). The emphasis will be on the presentation of a user's guide for a wide range of applications in many-body dynamics. We hope to convince the reader that the recursion method is a userfriendly yet powerful calculational tool. Technically, the core of the recursion method is the orthogonal expansion of a specific quantity. If the goal is to determine a time-dependent autocorrelation function for the solution I'l'(t» of the SchrMinger equation. In the Liouvillian representation, the equilibrium state of the system is specified, in general, by a stationary density operator. Depending on the application, that density operator may represent any eigenstate of the system or any suitable equilibrium ensemble. Here the specification of the state enters the recursion method in the form of an inner product that is needed for the execution of the orthogonal expansion of operator A. It is straightforward to transcribe the Liouvillian representation of the recursion method from the quantum formulation to the classical formulation. The dynamical variable A(t) is then a function in phase space. Replacing the quantum Liouvillian operator L u (commutator) by its classical counterpart L cl (Poisson bracket), transforms the iieisenberg equation into Hamilton's equation. The historical roots of the recursion method can be found in research areas as diverse as statistical mechanics, nuclear physics, solid state physics and applied

o

IThe recursion method has been used for other purposes not discussed in this work (Haydock 1980, Pettifor and Weaire 1985, Czycholl and Ponischowski 1988].

Section 1-2

5

mathematics, but for the types of applications emphasized in this work, it is squarely based on the projection operator formalism [Zwanzig 1961, Mori 1965]. Liouvillian representation

i"~ I'l'(t»

cIA = iLA dt

L qu

= .!.[H,

"

Hamiltonian representation

= H!'l'(t»

dt

], L cl = i{H, }

A(t) = eiLtA(O) 00

=L

k=O

I'l'(t»

=

e-iHtlh 1'l'(0» 00

Cit}fk

=:LDk(t)lf,;> k=O

f k+1 = iLfk - ...

lfk+1> = Hlf,;> - ...

f o = A(O)

lfo>

=

1'l'(0»



= A1

Table 1-1: Some characteristic quantities that distinguish the two main representations of

the recursion method. The Liouvillian representation will be further discussed in Chapter 3 and the Hamiltonian representation in Chapter 4.

Carrying out the program of the recursion method in the Liouvillian representation for the time evolution of a dynamical variable A(t) is equivalent to setting up and solving the generalized Langevin equation for that variable. This connection is especially useful for the interpretation of various approximation schemes commonly invoked in applications of the recursion method in terms of more or less familiar phenomenological models of dynamical systems on a contracted level of description (stochastic processes). The Hamiltonian representation of the recursion method grew out of attempts to solve an important problem in solid state physics - the determination of electronic densities of states under special circumstances, (i) situations in which band theory is unreliable, such as in strongly correlated systems, and (ii) situations in which band theory is inapplicable, such as in systems without spatial periodicity because of impurities, vacancies or amorphous structure. The local-environmentapproach to the electronic structure of solids [Haydock, Heine and Kelly 1972] introduced the recursion method to solid state theory as a powerful alternative to band theory.

6

Chapter 1

Expanding the Green's function which determines the desired density of states (or spectral density, in dynamics applications) into a continued fraction is equivalent to the tridiagonalization of the associated model Hamiltonian in a particular orthogonal basis. The latter task had already been a familiar one for many years [Lanczos 1950] and continues to play a significant role in the numerical diagonalization of large matrices. The body of mathematics that bears on the various implementations of the recursion method is quite extensive. Apart from the theory of continued fractions, we should also mention the close links to the classical moment problem and to the theory of orthogonal polynomials. How to reduce practically any problem to one dimension, was the title of the first talk (given by D.C. Mattis) at the 1980 International Conference on Physics in One Dimension. That title refers, of course, to the recursion method as an instrument by means of which a large set of possible problems in condensed matter physics can be reduced to a pseudo-ID problem, known as the chain model associated with the original problem. The chain model, characterized by the tridiagonal Hamiltonian produced by the recursion method, is isomorphic to a chain of local degrees of freedom with nearest-neighbor coupling only, "the archetypal normal mode problem in ID" [Mattis 1981]. This may tell us that ID model systems can be constructed to exhibit the most complex structures found in condensed-matter problems. But that is only of limited use as long as no isomorphism between the physical interpretations of such structures can be found.

1-3 Formalism and Goals Our presentation of the recursion method is preceded, in Chapter 2, by a brief account of the main results of linear response theory. This serves a twofold purpose: o It provides definitions for all the dynamical quantities that are of relevance in the context of the recursion method, and it clarifies some of their general properties. o It establishes the vital link between dynamics experiments and theoretical approaches (including the recursion method) to equilibrium dynamics under clearly defined assumptions (see Fig. 1-1). Having thus laid the foundation for both the formal development of the recursion method and the experimental relevance of its results, we shall proceed with the former in some detail in Chapters 3 to 9. The main lines of formal development are mapped out in Fig. 1-2. The Liouvillian representation of the recursion method will be introduced in Chapter 3 both in its quantum and its classical formulation (Secs. 3-1 to 3-4). The two main goals are the following: o We elucidate the physical meaning of the particular dynamical quantity which is the direct result of the recursion method: the relaxation function in the continued-fraction representation. o We provide a user's guide for the calculation of continued-fraction coefficients by the recursion method in that representation.

Section 1-3

7

An alternative method, which yields essentially the same data, is the moment method. In Sec. 3-5 we discuss how it is related to the recursion method.

Reconstruction of Spectral Densities

Figure 1·2: Overview of the main lines of formal development of the recursion method as presented in Chapters 3 to 9 for applications to many-body dynamics.

8

Chapter 1

The last three Sections of Chapter 3 branch out from the main line of formal development. In Sec. 3-6 we present a formal derivation· of the generalized Langevin equation for the dynamical problem at hand. The main benefit of that exercise will be drawn in Chapter 8, where phenomenolo$ical approximations of memory functions will be discussed. In Sec. 3-7 we turn our attention to the projection operator formalism, the original formulation of the recursion method. Our goal there is to deepen the understanding of what the physical consequences of an orthogonal expansion are and of how such a scheme could be motivated on grounds derived from physics rather than formal procedure. Section 3-8 introduces retarded Green's functions, principally for comparative studies (in Chapters 6 and 11) with the recursion method. In Chapter 4 the Hamiltonian representation of the recursion method is introduced along lines parallel to Chapter 3. The goal remains the same, but the recursive scheme proceeds via a different path and yields information on a different but related dynamical quantity (Secs. 4-1 to 4-3). For almost all dynamics applications of the recursion method in this representation, the equilibrium state of interest is the ground state of the system. Except for rare circumstances, the wave function of the exact ground state is not known analytically. It must be determined numerically for finite systems. Hence practically all applications of the recursion method in the Hamiltonian representation are preceded by the computation of the ground-state wave function. Recursive methods are recommended for that task. A survey of such methods including a comparative study of their performance is found in Secs. 4-4 to 4-8. The data that come out of the computational implementation of either representation of the recursion method are, in their most condensed form, a sequence of non-negative numbers {.!\}, ~, ... }. They are the continued-fraction coefficients of a certain relaxation function. That relaxation function determines the spectral density in whose properties we are primarily interested. These properties are, for example, the bandwidth iik is finite, the size of the gap if one is present, the singularity structure, and the detailed shape of the spectral-weight distribution. All this information is encoded in the .!\k-sequence, but only a finite number of coefficients is, in general, explicitly known. The retrieval of quantitative information on the above mentioned properties of spectral densities from incomplete .!\ksequences is clearly the most challenging task in typical applications of the recursion method. One popular but unsophisticated approach to this task is based on the use of an artificially truncated continued fraction combined with a more or less fancy way to convert a finite set of &-functions into a pseudo-eontinuous spectrum. We shall see that this type of analysis has little merit in many-body dynamics. However, one variant of this approach, the n-pole approximation, has played a significant role in the early development of the recursion method (Chapter 8). In this work we aim considerably higher than that. Our approach is based on a quantitative analysis (carried out in Chapter 5) of the finite .!\k-sequence (the data of the recursion method) prior to any attempt Of terminating the continued fraction. The key to any successful reconstruction of a spectral· density is that certain properties of the associated .!\k-sequence can be extracted with sufficient

9

Section 1-4

reliability from the limited number of coefficients that are known. We may call those properties the implicit information extractable from the sequence Ll!, Ll2 , ... , LlK in addition to the explicit information contained in the values of the first K coefficients. The amount of implicit information which can be inferred depends, of course, on the number of coefficients which are available and on the degree of complexity of the sequence under investigation. We shall see that under not too unfavorable circumstances it is possible to make quantitative predictions about bandwidths, gaps, singularity structures etc (Chapter 9). In addition to that we can determine the decay laws of the spectral density at high frequencies from the growth rates of the Llk-sequence. This decay law may serve as the basis for a definition of universality classes of dynamical behavior, a concept that will be discussed extensively in Chapter 7. The growth rate itself determines the convergence properties of the continued fraction and the analyticity properties of the spectral density. This is indispensible information for the construction of a termination function. The termination function will be constructed such that the completed continued fraction takes into account all the available explicit and implicit information from the known continued-fraction coefficients (Chapter 9).

1-4 Applications Almost all applications of the recursion method presented in this book will be in the general area of spin dynamics with emphasis on ID quantum spin models. Quantum spin chains are, in fact, physically realized in quasi-ID magnetic insulators for a large variety of spin models [Steiner, Villain and Windsor 1976; Willett, Gatteschi and Kahn 1985]. Typically these are compounds in which the magnetic ions are arranged in chains along one crystallographic axis such that the exchange interaction between ions within a chain is very much stronger than the interaction between ions belonging to different chains. Two of the most important experimental techniques which probe dynamical properties of quasi-ID magnetic insulators very directly are neutron scattering and NMR. Their connection to dynamic two-spin correlation functions are the following: o The inelastic scattering cross section for magnetic scattering of neutrons is directly proportional to the dynamic structure factor 2

d (J dQdro

o

oc

S ( 1.11.1

~

q,roj

=

-iqnf- dteiCiJt<SI.I(t)Sl.lI+n>'

1~ N LJ e I,n

I

(1 1) .

-00

In situations where a hydrogen atom is crystallographically located suitably close to the (electron) spin chain, the NMR spin-lattice relaxation rate liT! of this proton is dominated by the local fluctuations of the unpaired electrons of the nearest magnetic ion. It is proportional to the frequencydependent autocorrelation function at the nuclear Larmor frequency ~:

10

Chapter I +00

_1 Tt

cc

Sllll(~) = 2.E jdteioot.'<sj(t)sj> . N

I

(1.2)

-eo

Dynamics of 10 quantum spin models is the field of research where the authors of this work have been most active in recent years, which explains their choice of applications. However, the reader will find it straightforward to translate the methods of analysis designed for the applications presented here into equally powerful calculational tools within hislher own field of research. Applications of the recursion method will be presented on three levels of complexity and sophistication. On each level they serve a different purpose: o The simplest applications are to such toy models as the harmonic oscillator, linear spin waves, free fermions, classical and quantum spin clusters,... These applications (starting in Chapter 6) are used solely for didactic reasons, serving the purposes of (i) illustrating the "mechanics" of the recursion method and (ii) illuminating the links between the recursion method and other calculational techniques. o The first batch of applications to models with nontrivial spin dynamics will be presented in the first 9 Chapters, where the focus is on the development of the recursion method as a calculational technique. Only passing attention will be given to the underlying physics. o The bulk of applications to models of considerable interest in former and current research in spin dynamics will be presented in Chapters 10 and 11. Here the focus will be prinrarily on the physical phenomena discussed and less so on the methodological aspects. In these last two Chapters we shall portray the strengths and limitations of various general methods in the contexts of two major areas of research in spin dynamics. The theme of Chapter 10 is spin dynamics at high temperature. For all practical purposes, high temperature means T = 00 in this context. The main question of interest is an old one: What types of processes govern the transport of spin fluctuations in the absence of instantaneous spatial correlations? This question will be illuminated from different angles with results for classical and quantum spin models obtained by a number of calculational techniques including exact analysis, rigorous bounds, simulations, and the recursion method. Finally, in Chapter 11, we shall explore the fascinating world of lD quantum spin dynamics at zero temperature. At the center of attention will be the 10 s = 1/2 XYZ model and its special cases. We shall discuss the multi-faceted interrelations between the properties of (i) the ground state of the system, (ii) the excitation spectrum, and (iii) the spectral densities. These properties have been investigated by exact analysis, Green's function techniques in the fermion representation, field-theoretic approaches, finite-chain analysis, and the recursion method.

2 LINEAR RESPONSE AND EQUILffiRIUM DYNAMICS 2-1 Response Function and Generalized Susceptibility Dynamical properties of a many-body system specified by a Hamiltonian Ho in thermal equilibrium are most frequently expressed in terms of time-dependent correlation functions of the form (t) == {

(2.32b)

2'l\2k

2'l\2k+l

in terms of 2k or 2k+l commutators and an anticommutator or an additional commutator. The direct reconstruction of the functions S(oo), «1>(00), or X"(oo) from the moments Mn is well known to be an ill-conditioned problem, in general, and worse if the Mn grow faster than n! for large n; in that case the above mentioned functions are no longer uniquely determined by their moments (more about that in Sec. 5-4). We shall see that many of the practical difficulties which plague the moment method can be circumnavigated by the recursion method in combination with a continued-fraction analysis.

3 LIOUVILLIAN REPRESENTATION Liouvillian dynamics provides a direct link between classical and quantum mechanics. It provides the language for a direct transcription of the recursion method from a quantum formulation to a classical formulation. At the same time, it is the basis of a particular representation of the recursion method, which we call the Liouvillian representation. A strikingly elegant and user-friendly formulation of this representation has resulted from Lee's important developmental work [Lee 1982, 1983].

3-1 Quantum Formulation The dynamical system is specified by a quantum Hamiltonian H and a dynamical variable (operator) A in whose time evolution we are interested. The goal is to determine the dynamic correlation function -

,

(3.5)

Po where = Tr[e-~HA]/Z, Z = Tr[e-~H], P= l/kBT. It was designed such that the

Fourier transform of the associated fluctuation function (A(t),A(O» satisfies the classical fluctuation-dissipation theorem (2.25) also for quantum systems at arbitrary

INote that we have set h=1 everywhere except in Chapters 1 and 2.

18

Chapter 3

temperatures. The quantity ~(A,B) is then a static susceptibility. A more general scalar product has the form (A,B)

1~

= _jdA.g(A.)<eAHA te-AHB> ~o

-

,

(3.6)

with the weight function g(A.) satisfying the conditions g(A.) ;;:: 0,

g(~ -A.) = g(A.)

1~

,

- jdA.g(A.)

~o

=1

(3.7)

The Kubo inner product (3.5) corresponds to a constant weight function, g(A.)=1. An important alternative choice is based'on the weight function

1 = _~[O(A.) +O(~ -A.)]

g(A.)

2

.

(3.8)

The resulting inner product, (A,B)

= .!.. 2

- ,

(3.9)

is more readily evaluated than (3.5) at ~>O including the important case of zero temperature. The recursion method with this inner product yields the standard spectral density (0), which satisfies the general fluctuation-dissipation theorem (2.24). In the infinite-temperature limit (~~), the inner product (3.6) is the same for bQth choices of g(A.) : (A,B)

=

- .

(3.10)

3-2 Classical Formulation In the classical case, the dynamical system is specified by a Hamiltonian function H(ql ,···,qn;Pl,·..,Pn) = const and a dynamical variable A(ql ,...,qn;Pl ,... ,Pn) = A(t). The time evolution of A(t) is determined by Hamilton's equation of motion,

_dA = -{H,A} = iLA dt

,

(3.11)

where

L= i{H, }= it(aH ~ _aH~) j=l

aqj apj

(3.12)

apj aqj

is the classical Liouvillian operator (again Hermitian), now operating on classical dynamical variables. The canonical coordinates qj' Pj are special dynamical variables which satisfy the fundamental Poisson brackets

Section 3-3

{qj,qj} = {Pj'p) = 0

, {qj'Pj} = Ojj .

19

(3.13)

For the repeated evaluation of Poisson brackets {H, } as required by the recursion method, it is sometimes more convenient to express the energy function H and the dynamical variable A in terms of a set of elementary but non-canonical dynamical variables ul'''''um , if the Poisson brackets of these variables {uj'Uj } = Pij(ul"",u m )

(3.14)

have a sufficiently simple structure. The inner product which produces the classical fluctuation function as defined in Chapter 2 is the following : (A,B)

=

- ,

(3.15)

where the expectation value denotes the phase-space average

= ~Jdnqdnpe-~H(q,p)A(q,P) .

(3.16)

It can be interpreted as the classical limit of the general quantum inner product (3.6), Le. the limit in which the operators A, B and H turn into classical functions of the phase-space coordinates. 3-3 Orthogonal Expansion of Dynamical Variables At the core of the Liouvillian representation of the recursion method is the orthogonal expansion of the (quantum or classical) dynamical variable under scrutiny [Lee 1982]: A(t) =

L Ck(t)fk .

(3.17)

k=O

For a classical system, the A are an orthogonal set of functions in phase space, for a quantum system an orthogonal set of operators. In both cases they span a Hilbert space of (generally) infinite dimensionality. The Liouville operator acts on the vectors A (functions or operators) in this Hilbert space. The orthogonal expansion (3.17) is then carried out in two steps: #1 Determine a particular orthogonal basis {fk} in the Hilbert space of dynamical variables by applying the Gram-Schmidt procedure with the Liouvillian L as the generator of new directions. #2 Insert the expansion (3.17) into the equation of motion - the Heisenberg equation (3.1) for quantum systems or Hamilton's equation (3.11) for classical systems - to obtain a set of differential equations for the timedependent coefficients Ck(t). For step #1 we note that the general inner product (3.6) between the vectors A and iLA vanishes for arbitrary A:

20

Chapter 3

(A,iIA)

=0 .

(3.18)

This simplifies the Gram-Schmidt orthogonalization considerably and leads to the following set of recurrence relations for the vectors A:

A+I

(3.l9a)

= iLlk + ~k!k-I' k =0,1,2, ...

~k =

(/k,fk) , k=I,2,... (/k-I,fk-I)

(3.l9b)

with f I = 0 and 10 =A. The first three iterations are illustrated in Table 3-1. We shall see that the sequence of numbers ~k contains all the information necessary for the reconstruction of the fluctuation function (A(t),A(O)) or the associated spectral density.

10 "

A

(initial condition)

ilfo .. 11 -. (/0/1) .. (/o,ilfO>. .. 0

ilfl .. 12 - Al/o -. (/lJ~ .. (/1,ilfl)+AI(/IJO> .. 0 -. (/oJ~ .. (/0.ilf1) +A1(/oJO>

if

.. -(ilfOJ 1) +AtifoJO> .. 0

A (/IJI) I .. (/oJO>

ilf2 .. 13 -A,/I - r ,to -. (/2J3) .. (/2.ilf~ +~(/2JI) +r2(/2JO> .. 0 -. (/lJ~ .. (/1.ilf~+~(/lJl)+r2(/IJO> • -(ilf1J 2) +Az(/I J 1) .. 0

if Az .. (/2J2) (/lJI)

-. (/oJ~ • (/o.ilf~ +~(/OJI) +r2(/oJo>

.. (/IJ2)+r2(/OJO> .. 0

if r 2 .. 0

.Table 3-1: The first three iterations of the orthogonalization procedure carried out in detail.

In step #2 we insert the orthogonal expansion (3.17) into the equation of motion (3.1) or (3.11). The differential operator acts on the Ck(t), and the Liouvillian acts on the A with the result given by (3.19). Term-by-term comparison of the coefficients of each vector Ik in the equation of motion then yields the following set of coupled linear differential equations for the functions Ck(t): Ck(t) = Ck_l(t) - ~k+lCk+l(t),

k=0,1,2,...

(3.20)

Section 3-4

21

with C_1(t) == 0, Ck(O) = 0k,O' Unlike the vectors!k in (3.19), the Cit) cannot be detennined recursively. If our goal is to detennine the fluctuation function of the dynamical variable A(t), then it suffices to know just one of the functions Ck(t):

C (t)

o

=

(A(t) ,A(O» (A(O) ,A(O»

=

«I>(t) .

«1>(0)

(3.21)

This relation follows directly from the orthogonal expansion (3.17). Throughout this book, the inner product (3.9) will be used for quantum systems and (3.15) for classical systems. The last expression in (3.21) is the (normalized) fluctuation function with «I>(t) as defined in Chapter 2.

3-4 Relaxation Function and Spectral Density A formal solution of the coupled differential equations (3.20) can be obtained by Laplace transform. The functions (3.22)

Ciz) == jdte-ztCk(t)

o satisfy the set of algebraic equations zciz) - 0k,O

= ck-I(Z)

- L\k+lck+l(z),

k=0,1,2,...

(3.23)

with c_ 1(z) == O. In order to solve these equations, we rewrite (3.23) for k=O in the form co(z)

1

= ---~ C1(z) z + lll-A

(3.24)

co(z)

and express the ratio ck(z)lck_I(Z) in the denominator recursively (for k=1,2,...) in terms of ck+I(Z)lck(z), again from (3.23). The result is the function co(z) in the continued-fraction representation: co(z)

1

= ---.....,....-L\l

Z + ----,..-

(3.25)

~

z+-Z + ...

The functions cI(Z), c 2(z), ... can then be determined recursively from (3.23), once co(z) has been detennined. The result (3.25) demonstrates the claim made earlier that the L\k-sequence alone fully determines the correlation function under investigation. The function co(z) is, ,named relaxation function. It represents the Laplace transform of the normalized fluctuation function (3.21). The latter is thus recovered by the inverse Laplace transform of (3.25),

22

Chapter 3

Co(t)

= ~ jdzez1co(z) 21tz c

,

(3.26)

where the integral is performed along the path C parallel to the imaginary axis in the complex z-plane as depicted in Fig. 3-1. In most applications, we wish to determine the Fourier transform of Co(t) rather than the function Co(t) itself. The normalized spectral density,

J

"-

=M2n result from the relations,

(n-l) _ M2k -

A

"""n-I

M(n) 2k

An-I M (n-Z) + - - 2k-Z '

(3.34)

An - Z

for n=k,k-l,...,l and k=1,2,...,K and with set values Mtl)=O.

MZ = Al

M4

M6

Al = Mz

M

= Al (AI +Llz)

4 Az = _-M z Mz

= Al [(AI +Az)z+LlzA3l

A3 =

M6IMz -M4 M4 MiMz-Mz Mz

Table 3-2: The first 3 expansion coefficients of (3.32) expressed in terms of the first 3 continued-fraction coefficients of (3.25) and vice versa.

These simple transformation relations establish the link between the recursion method and the various techniques in use for the analysis of moment expansions. The computational effort of calculating the moments (3.31) is similar to that of calculating the same number of Ak's by the recursion method. However, we shall see that working with continued-fraction coefficients has at least two advantages: (i) the Ak's are more suitable for the direct reconstruction of the spectral density (1)0«(0); (ii) information on specific properties of the spectral density, such as bandwidths, gap sizes, edge singularities, infrared singularities, and decay laws at high frequencies can be extracted directly from the Ak-sequence.

3-6 Generalized Langevin Equation In the context of transport theory, the formalism of the recursion m~thod may serve as a link between microscopic model systems and phenomenological theories bas~ on a contracted level of description. The classical relaxator, specified by the

Langevin equation, A(t) +

~(t) = F(t) 't

,

(3.35)

is a prototypical example. In this simple system, the quantity described by the dynamical variable A is subject to a linear damping, specified by the relaxation time

Section 3-6

25

't, and by a B-correlated random force F(t) (white noise). The resulting fluctuation function is then a pure eXRonential, CO(t)=e- t/t , and the spectral density a Lorentzian, o(00)=2't/(1 +002't ). Now consider the 'generalized Langevin equation, t

A(t) + JdtlI;(t-tl)A(t l ) = F(t) .

(3.36)

o In the context of a phenomenological theory, the memory function I;(t) contains essentially the entire model specification. The associated random force F(t) cannot be chosen freely but must satisfy the consistency c6nditions imposed by the fluctuation-dissipation theorem. Remarkably, the phenomenologically motivated generalized Langevin equation provides, under very general circumstances, a rigorous description of quantum or classical many-body systems. For any given microscopic (quantum or classical) Hamiltonian model system, the generalized Langevin equation for the dynamical variable of our choice can be derived within the formalism of the recursion method [Dupuis 1967, Lee 1983]. Let us define the two functions, c1(z) L(Z) ;: A1- co(z)

,

ck(z)

biz);: - _ , k=I,2, ... co(z)

(3.37)

where co(z) can be determined from first principles in the continued-fraction representation (3.25), and the ck(Z) for k~1 can be inferred recursively from Eqs. (3.23). We rewrite these latter equations in the form ZCO(Z) - 1 + L (z)co(z)

=0 ,

(3.38a) (3.38b)

The inverse Laplace transforms of these functions then satisfy the equations t

to(t) + JdtlI;(t-tl)Co(tl) = 0 ,

(3.39a)

o t

tk(t) + JdtlI;(t-tl)Ck(t l )

= Bk(t),

k= 1,2,...

(3.39b)

o From Eqs. (3.39) and the expansion (3.17), we obtain the generalized Langevin equation (3.36) for the dynamical variable A(t). In this context, the memory function I;(t) and the random force F(t) are expressed in terms of microscopic properties of the model. However, we must keep in mind that both Ut) and F(t) are not explicitly known prior to the solution of the dynamical problem. The functions Bit), k=1,2, ... can be interpreted as the coefficients of an orthogonal expansion of the random force,

26

Chapter 3

F(t)

= E Bit)fk

, Bk(O)

k:}

= ak,}

,

(3.40)

in the same basis lftl as previously used for the dynamical variable. Note that F(t) has zero projection onto f o' which guarantees that there is no correlation between the dynamical variable at time zero and the random force at any later time: (F(t),A(O)) = O. In phenomenological theories, the random force is usually specified by its fluctuation function. The latter is equal to the first coefficient of the orthogonal expansion (3.40): (F(t) ,F(O)) (F(O) , F(O))

(3.41)

3-7 Projection Operator Formalism Having introduced up front a modern and user-friendly formulation of the recursion method, we now wish to have a closer look at the way it was originally designed by Zwanzig [1961] and Mori [1965]. This earliest formulation is known under the name projection operator formalism or memory function formalism. Today it can be regarded as a particular formulation of the recursion method in the Liouvillian representation. From a computational point of view, it is an unwieldy tool that has discouraged many a potential user through the years. Its main attractive feature is that it provides a physical motivation for the orthogonal expansion which is at the basis of any variant of the recursion method. It was the hope of the inventors of the memory function formalism that by means of a few orthogonal projections the most essential features of the overwhelmingly complex many-body dynamics could be separated from the rest. That hope remained unfulfilled in all but the simplest applications. However, the pioneering effort led the way to more successful methods of. analysis and modes of representation. Following Forster's insightful account [Forster 1975] of the memory function formalism, we represent the (classical or quantum) dynamical variables in terms of bras, , and identify the brackets with the inner product (A,B) introduced in Secs. 3-1 and 3-2. Likewise, matrix elements of any Hermitian operator such as the Liouvillian L have an obvious interpretation: == (A,LB) = (LA, B). In this notation, the dynamical quantity of interest the (normalized) fluctuation function for the variable A(t) - reads C (t)

o

= =

= IA(O» == Ifo> as in Sec. 3-3, the first projection operator to be used and its complement are

Po ==

I

lfo>-- and a new Liouvillian L,. The operator L, does no longer contain the full many-body dynamics. It is obtained from the original Liouvillian by having it sandwiched between a pair of projection operators Qo' These projection operators act like filters; they absorb that part of the full many-body dynamics which has been dealt with explicitly in the course of this first projection. The explicit information is contained in the normalization constant of the memory function I:, (z) as we shall see. Carrying out another projection means repeating the calculational steps that led from (3.46) to (3.52) for the new; slightly reduced dynamical problem. The result is the old memory function in the role of anew (still not normalized) relaxation function

I: ,(z)

(3.53)

z +I:lz)

expressed in terms of its own memory function

I: 2(z)

1

1



z+ IL2

= --

(3.54)

with ~, = /, !f2> = iQ,L,!f1>, Lz = Q,L,Q,. Repeating this cycle over and over again produces a continued fraction which, after n iterations, is terminated by the nth-level memory function I:n(z): 1

co(z) = - - - - - - ; - - - - ~,

Z

+-------z

+ ...

(3.55)

Section 3-8

29

The coefficients {Ll l , ..., ~-d reflect the explicit dynamical information extracted by means of the first n projections. The last coefficient, Lln , is part of the function ~n(z).

Each iteration adds a layer of projection operators around the original Liouvillian. If n is sufficiently large, so it has been argued, all distinctive spectral properties of L will be filtered out and incorporated explicitly into the relaxation function via the continued-fraction coeficients Llk • What remains of L is then believed to be indistinguishable from a source of white noise. The associated memory function ~n(z) would then be a constant, equal to the inverse relaxation time l/'tn of a classical relaxator process, a process described by. the simple Langevin equation (3.35). It turns out that this type of reasoning is not really valid except for special circumstances. The best known and most frequently quoted example is the relaxation function associated with the density fluctuations in a normal fluid as can be probed by light scattering [Forster 1975]. These fluctuations are governed by two processes - heat diffusion and sound waves - giving rise to the well-known three-peak structure in the corresponding spectral density (the Rayleigh peak at ro=O and one Brillouin peak on either side). These structures are recovered by the projection operator formalism after one or two projections, respectively. We shall return to that theme in Chapter 8, in the context of our discussion of the n-pole approximation.

3-8 Retarded Green's Functions In the context of this book, the Green's function method can be regarded as another general method for the calculation of dynamic correlation functions of many-body systems [Elk and Gasser 1979, Lovesey 1980, Doniach and Sondheimer 1974]. As in the case of the recursion method, the application of the Green's function method to nontrivial models poses challenging tasks of approximation and interpretation. It is not a priori clear which of the two general methods is likely to produce a more reliable result in a given application where neither method can be carried through without resort to some sort of approximation. A number of comparative applications will be presented later for the twofold purpose of (i) testing their performance under specific circumstances or (ii) gaining complementary pieces of information on a complicated dynamical problem. Our goal in this Section is to establish the formal connections between the fundamentals of the Green's function formalism and the Liouvillian representliltion of the recursion method. We begin with the definition of the two varieties of the retarded Green's function, G±(t-t') :; -

(3.57)

.

The retarded Green's function satisfies the equation of motion, i ;t±,

(3.58)

which is derived from (3.56) and the Heisenberg equation (3.1) for the operator A(t). The equation of motion for the frequency-dependent Green's function +00

(3.59) is then algebraic in nature, ± m(00)

= -lim S[G+(oo+ie)] £ ..... 0

,

X" (00) = -limS[G_(oo+ie)]. (3.63) £ ..... 0

The connection to the Liouvillian representation of the recursion method is then established by the following relation between the relaxation function (3.25) or (3.43) and the Green's function G+(I;) for A=B:

Section 3-8

G+(~) = -2i$(t=O)cO(Z) = -2i z+IL

,

31

(3.64)

where ~=iz, and $(t=0) = is the initial value of the fluctuation function, expressed in tenns of the inner product as used in Sec. 3-7.

4 HAMILTONIAN REPRESENTATION This alternative representation of the recursion method is designed for the study of dynamic correlation functions of a quantum Hamiltonian system in its ground state.} An important initial task in most practical applications is therefore the determination of the ground-state wave function ICPO> of the system. There exist quite a few computational methods for that purpose. Some of them are based on the Lanczos algorithm for the tridiagonalization of large matrices [Lanczos 1950]. That algorithm can, in fact, be regarded as one of the roots from which the recursion method has originated. The connection will be established and discussed in Secs. 4-4 and 4-9. Two widely used recursive algorithms for the determination of groundstate wave functions - the modified Lanczos method and the conjugate-gradient method - will be presented in Secs. 4-5 to 4-8 along with a comparative performance test. In the meantime, for our description of the Hamiltonian representation of the recursion method, the assumption is that we know the groundstate wave function for a system of given size.

4-1 Orthogonal Expansion of Wave Functions If our goal is to determine, for a given quantum Hamiltonian H and its ground-state wave function Icpo>, the normalized correlation function,



So(t) = ---:-~,..,..-

(4.1)

of the dynamical variable represented by the Hermitian operator A, then we can accomplish that task along two different avenues. We either determine the fluctuation function (3.21) via the Liouvillian representation of the recursion method and then convert the result into the correlati?n function using the fluctuation-dissipation theorem, or we determine the correlation function (4.1) directly via the Hamiltonian representation of the recursion method as described in the following [Gagliano and Balseiro 1988]. In order to simplify the expressions for the dynamical quantities as produced in the Hamiltonian representation, we consider henceforth the Hamiltonian ii = H-Eo' whose ground-state energy has been shifted to zero. The time evolution of the wave function,

",,(t»

= A( -t)lcpo> ,

(4.2)

is governed by the Schrodinger equation,

}Any other eigenstate of the Hamiltonian would be acceptable too, but for stationary states that do not correspond to thermal equilibrium some properties of dynamic correlation functions are different from those discussed in Chapter 2.

33

Section 4-1

d

i_I'I'(t» dt

= H1'I'(t» .

(4.3)

The recursion method is based on an orthogonal expansion of that quantity:

1'I'(t»

= L Dk(t)!fy ,

(4.4)

k=O

where {!fy} is an orthogonal basis in the Hilbert space of the Hamiltonian operator H. Carrying out the expansion involves again two steps: #1 Determine the orthogonal basis {!fy} in the Hilbert space of wave functions via Gram-Schmidt with initial condition !fo> = A1o> and with the Hamiltonian as the generator of new directions. #2 Insert the expansion (4.4) into the equation of motion - here the SchrOdinger equation (4.3) - to obtain a set of differential equations for the timedependent coefficients Dk(t).

lto>

= AI~o>

It, >

=

Hlto>-aolto>

lt2>

=

HI!,> -a,lt,> -b,Yo>

(initial condition)

-+



-+

-ao

=



-

lt3>

=

2

-

2

= -a, -b,

Hlt2>- a2lt2>-b2Y,> - c2lto>



= -a, -b,

= -b,2 -+

ao =

if

=0

=0

if

2 b , -

=0

if

a, =

-+



-+

= 0

if

b2

-+

= 0

if

~=

=0



if c2 = 0

2



=--



Table 4-1: The first three iterations of the orthogonalization process generated by the Hamiltonian H.

The first three iterations of step #1 are illustrated in Table 4-1. Successive orthogonal vectors !ft> are produced by the following set of recurrence relations:

34

Chapter 4

(4.5 a)

(4.5b)

2

bk

=

= ,

(4.5c)

k=1,2,...,



=

with !f. 1> 0 and !fo> A1cJlo>. It will turn out that the double sequence of numbers {ak' bi} is all that is needed for the reconstruction of the dynamical quantity of interest, the correlation function S(t). In step #2 the Schrodinger equation (4.3) is applied to the orthogonal expansion (4.4). The differential operator acts on the Dk(t) and the Hamiltonian on the !f~. The result is the following set of coupled linear differential equations for the functions Dit): iD k(t) = D k_1(t) +

a~k(t)

k =0,1,2,...

+ bk2+1Dk+l(t),

(4.6)

with D_ 1(t) == 0, Dk(O) = ~k.O' The first one of these functions is equal to the (normalized) correlation function we set out to determine: 2



_

S(t) _ = So(t) .

- -

S(O)

(4.7)

4-2 Structure Function The differential equations (4.6) are converted by Lap1ace. transform, dk(S)

f

= dtei~tDit)

(4.8)

,

o

into algebraic equations

(S -ak)dk(s)

-

i~k.O = d k- 1(S)

+ b;+ldk+l(S),

k=0,1,2,...

(4.9)

with d_ 1(S) == O. This set of equations can be solved for do(S) in the continued-fraction re~resentation by manipulations similar to those used in Sec. 3-4 for expression (3.25):

2Except for a slight mismatch in definition if ># O. 3We shall refer to do(~) by the name relaxation function, a term also used for the function Co(z) in Sec. 3-4. There is little danger of confusion in any given context.

Section 4-3

35

i

do(~)

2

b1

~ - ao -

(4.10)

2

b2

~ - a1 -

~ - a2

-

The normalized structure function, +00

(4.11) is then recovered directly from (4.10) via the relation So(oo)

= lim 2~[do(oo+iE)]

(4.12)

£-+0

4-3 Continued-Fraction Coefficients and Frequency Moments We recall from Chapter 2 that the structure function So( (0) is real (and equal to zero for ro0(00), into the continued-fraction coefficients !J.k of the function co(z), and vice versa. If we wish for the to convert the double sequence of continued-fraction coefficients ak' function do(~) as produced by the Hamiltonian representation of the recursion method into a single sequence !J.k for use in the continued-fraction analysis proposed later, then we need another algorithm - one for the conversion of the ak' into the M n and vice versa. We have derived such an algorithm from equations (4.9) applied to the asymptotic expansion

bi

bi

do(~)

= iL Mn~-(n+l)

,

(4.14)

n=O

which is obtained from the power series (4.13) by Laplace transform. The result is a set of recurrence relations, similar to (3.33) and (3.34), between the continued-

36

Chapter 4

bi

fraction coefficients ak' and the moments M n . These relations are most conveniently expressed in terms of two arrays of auxiliary quantities L k(n) and M k(n): o Forward direction: Given a set of moments Mo=l, M 1,...,M2K+I' the continued-fraction coefficients aO, ...,aK and bI,...,bi are obtained by initializing Mk(O) = (-l)kM

k'

for k

L

(0)

k

= (-l)k+1 M

k+1 '

(4.l5a)

= O, ... ,2K and then applying the recurrence relations (n-1) M(n) _ L(n-1) _ L(n-l)Mk k

-

k

--C;;:O' M _

n-I

(4.l5b)

n 1

L (n) le

M(n-I)

=

k

M(n-I) n-I

=

for k n,...,2K-n+l (in two successive inner loops) and n loop). The resulting continued-fraction coefficients are b:

o

= M~n),

an

= _L~n) "

n=O,...,K.

(4.l5c)

= 1,... ,2K (outer (4.l5d)

Reverse direction: Now the proper initialization is 2 M n(n) = bn'

L(n) = -a n n'

=0 K n- ,..., ,

(4.l6a)

(where b~ = b:1 == 1), and Mt1)

= 0,

k=O, ...,2K+l.

(4.l6b)

The recurrence relations to be carried out for n = O,... ,min(K,2K-J) (inner loop) andj = O,... ,2K+l (outer loop) are 2

bn (n-I) (n) 2 (n) M n+j+1 =bL. n n+J + -2-M n+j ,

(4.l6c)

bn - I L (n) _ M(n+l) an M(n) n+j+1 n+j+l - 2 n+j+l

(4.16 = Hl/o>-aolfQ>

1f2>

0

b2 a2 b3 ...

0

0

b3 a3

(4.18)

...

= -2ao+a; d . -+ - = 0 If ao = --;:-;-::-_ ao k~> -+ = 0 -+

= HI/I> -allfl > -bIYO> 2 -+ = +a I +b: -2a I -2b[ d da l

-+ - -+

=0

~ =0 2 db 1

. If a l

=....:.,..,..,....:;...

if b[ =

Table 4-4: The first three iterations of Lanczos' original scheme of tridiagonalizing a Hamiltonian matrix H. For given initial vector Ifo>' the algorithm produces the matrix elements {sk' b/} of (4.18) along with the orthogonal but not yet normalized basis {If?}.

The main advantage of the Lanczos tridiagonalization over other methods which accomplish the same task, such as the Householder method, is its very low

40

Chapter 4

memory requirements. The input consists of the first basis vector luo> and the operator rules for the recurrence relations (4.17). The orthonormal basis {Iu~} is generated along with the matrix elements {a", bi} and needs to be stored only if necessary for further calculations. Once the Hamiltonian has been brought into tridiagonal form, its eigenvalues can be determined by standard methods. For the computation of one or several low-lying eigenstates, the bisection method (for eigenvalues) combined with the method of inverse iterations (for eigenvectors) has proven to be very useful [Cullum and Willoughby 1985, Nishimori and Taguchi 1986]. In the context of the recursion method, however, the main significance of the Lanczos algorithm is a different one. Step #1 in the Hamiltonian representation is nothing but the Lanczos algorithm slightly disguised. The difference between the two sets of recurrence relations (4.5) and (4.17) (see also Tables 4-1 and 4-3) is that the orthogonal basis of the former is not normalized. We have lu~ = IA>l = Alcj>o> rather than for the determination of eigenstates. The relationship between tridiagonal Hamiltonian matrices and Green's functions in the continuedfraction representation will be further discussed in Sec. 4-8. Finally, we should like to point out that Lanczos' original formulation of his in the algorithm invokes a minimization condition for the coefficients a", recurrence relations (4.5) rather than the orthogonality condition used in Table 4-1. The two conditions are obviously equivalent. In Table 4-4, we display the first three steps of the Lanczos scheme as based on minimization criteria. In Secs. 4-5 to 4-7 we report a comparative study of algorithms (also based on minimization principles) for the computation of ground-state energy eigenvalues and eigenvectors of large Hamiltonian matrices [Nightingale, Viswanath, and Muller 1993].

bi

4-5 Modified Lanczos Method In an effort to adapt the standard Lanczos algorithm for a more direct computation of the ground-state energy and wave function of a Hamiltonian system without the intermediate step of tridiagonalization, a modified version which accomplishes precisely that was developed by Dagotto and Moreo [1985]. The modified Lanczos method, which happens to be a special case of the iterative Lanczos method discussed much earlier by Karush [1951], was first applied in a lattice-gauge-theory context and later introduced by Gagliano et al. [1986] to condensed matter applications. It has since been widely employed for the study of ID and 2D quantum spin models and models of strongly correlated electronic systems. The idea behind the method is to embed the standard Lanczos recursive cycle within another recursive cycle. The outer cycle terminates the inner one after one iteration and resets the initial condition. In practice, the two cycles make up a single loop consisting of two iterative steps. The loop is started by an initial step and terminated by a user supplied convergence criterion. Initial step: Select a (normalized) trial vector 1'1'0> for the ground state of the system, which is specified by a Hamiltonian H. 1'1'0> must have a nonzero

Section 4-5

41

projection onto the true (but unknown) ground-state wave function 14>0>. Different choices of starting vectors 1'1'0> will result in different rates of convergence. Iterative step #1: Given the kfh approximate vector l'Py . apply one cycle of the standard Lanczos algorithm to generate a vector Iyy which is orthonormal to 1'1'y:

'yy =

(H -k)I'I' y

,

(4.19)

2

Vk-i where in the subspace spanned by l'Py and Iyy such that it minimizes the energy. Here it is obtained by diagonalizing the 2x2 matrix with elements

(4.20)

The lower eigenvalue reads (4.21) where £k

= k and 3

2-

3

k- 3kk +2k

2(. 1'1'1>' ... converges toward the exact ground-state wave function 14>0>. We recommend the use of the following convergence criterion: 2

2

k-k - - - - :2 - - < k where £ is comparable to machine precision.

£,

(4.24)

42

Chapter 4

In the modified Lanczos algorithm, memory must be allocated for the simultaneous storage of three vectors, l'Py , H1'P y , H21'Py , in the kth iteration. Each iteration involves two matrix multiplications.

4-6 Conjugate-Gradient Method The conjugate-gradient method has long been known in the context of minimizing functions of several variables [Hestenes and StiefeI1952]. It was designed such that for quadratic functions in n variables the algorithm is guaranteed to converge after n steps. That property has the effect that even for more general functions the rate of convergence is typically enhanced considerably as compared, for example, to the steepest-descent (SD) method. 4 In the context of an eigenvalue problem HIx> = E1x>, the same approach can be taken for the minimization of the Rayleigh quotient R <xIHIx>/<xIx>, as was first suggested and demonstrated by Bradbury and Fletcher [1966]. The fact is that the minimum (maximum) value of R is equal to the lowest (highest) eigenvalue of H. The conjugate-gradient method represents an efficient recursive algorithm for the minimization of the Rayleigh quotient. It has proven to be a reliable computational tool in statistical mechanics, notably in the context of the transfer operator approach [Nightingale 1990]. Its algorithm is formulated here, like the modified Lanczos method, in terms of an initial step followed by two iterative steps forming a loop that is terminated by a user supplied convergence criterion [Nightingale, Viswanath, and Muller 1993]. Initial step: Select a trial vector Ixo> (not necessarily normalized) with nonzero projection onto the ground state wave function 10>. Iterative step #1: Given the ~ approximate vector Ixy , apply the gradient to the Rayleigh quotient

=

R _ <xklHIxy k - <xklx y to generate a vector Ig y which is orthogonal to Ix y

Ig y == VR k

=

2

<xklxy

(4.25)

:

[HIxy-Rklx y ]

(4.26)

Iterative step #2: Construct a new vector of the form Ixk+1>

= Ix y

+ (J,k1py

,

(4.27)

where (4.28)

40etailed discussions of the conjugate-gradient and steepest-descent methods in that context are found, for example, in Golub and Van Loan [1983] and in Press et al. [1986].

Section 4-7

43

and uk_1 = l for k~l (u_ I = 0), by minimizing the Rayleigh quotient Rk+ I of (4.27) with respect to the real parameter a.k. The condition dRk+I/dClk = 0 leads to the quadratic equation (4.29) with coefficients (4.30a) (4.30b) (4.3Oc) The larger one of the two solutions of (4.29) minimizes R k+ l • Termination: The sequence RI' Rz' ... of minimized Rayleigh quotients converges toward the exact lowest eigenvalue Eo of H, and the sequence Ix l >, Ixz>, ... of vectors converges (after normalization) toward the corresponding eigenvector 10>. The convergence criterion corresponding to (4.24) reads <Xklx J?>

----=--Z Rk

< 4£ .

(4.31)

Note that the vector IpJ?> (unlike IgJ?» is in general not orthogonal to 1xJ?>. The second term in (4.28), uk_IIPk_I>' has the effect of stabilizing the direction of the path in the Hilbert space toward the minimum of the Rayleigh quotient. This enhances the rate of convergence as will be demonstrated later. What is the most economical implementation of the conjugate-gradient method? The answer depends on whether the most valuable (or most limited) resource is (a) available CPU time or (b) accessible memory. Implementation (a) requires only a single matrix multiplication per iteration (one less than modified Lanczos) but needs memory for the simultaneous storage of four vectors (one more than modified Lanczos). Implementation (b), by contrast, requires two matrix multiplications per iteration and storage space for three vectors, just like the modified Lanczos method does. The sequence of computations in one iteration of the two implementations of the conjugate-gradient method as dictated by the above mentioned minimum requirements are summarized in Table 4-5.

4-7 Steepest-Descent Method The steepest-descent method is based on the same principle as the conjugategradient method - the minimization of the Rayleigh quotient for a one-parameter vector. But unlike the latter the former constructs the new vector Ixk+ l > from the previous one and the gradient of its Rayleigh quotient R k alone, (4.32)

44

Chapter 4

Results of (k_l)'h iteration:

0 Ixt>, Hlxt>, lPt.l>' <x!Jxt>, <x,)HIx,,>, =(2/<x!Jxt>)[Hlxy-R"Ixt>]

l(b)

vector addition: Igt>=(2!<x!Jxt»[Hlxt>-R!Jxt>] , overwrites Hlxt>

2

inner product: , u"_l

3

vector addition: IPt>=-lgt> + u"-llp"-l>

4(a)

inner products: <x"IPt>, , ,

5

matrix multiplication: Hlpt> ' overwrites Igt>

6(a)

inner product:

6(b)

inner products: <x~HlPt>,



7

quadratic equation: <X"

8

vector addition: Ix"+l>=lxt> + <X~Pt>, overwrites Ixt>

-

9(a)

vector addition: HIx"+l>=Hlxt> + <xtlflpt> • overwrites Hlxt>

9(b)

matrix multiplication: HIx"+l> • overwrites Hlpt>

10

inner products: <x"+11x"+1>' <x"+lIH1x"+l>' R"+l

Table 4·5: Sequence of computations to be performed during the J(h iteration of the conjugate-gradient method. Implementation (a) involves one matrix multiplication and requires memory for four vectors; implementation (b) involves one additional matrix multiplication but requires memory for only three vectors. The same sequence of computations, but with several simplifications, applies to the steepest-descent method discussed in Sec. 4-7 [from Nightingale, Viswanath, and MOller 1993].

and thus ignores the directional information from the previous iterations that is contained in the vector IPt> used in the conjugate-gradient method. Nevertheless,

Section 4-8

45

the general structure of the algorithm remains the same as in the conjugate-gradient case. Implementations (a) and (b) detailed in Table 4-5 are still applicable, but with some obvious simplifications. None of them reduces the number of matrix multiplications per iteration or the number of vectors to be stored simultaneously. It is fairly obvious that the modified Lanczos and steepest-descent methods are equivalent. For Ipy =-Igy , the coefficients (4.30) of the quadratic equation that determines the parameter (J,k in (4.32) can be expressed in terms of <X~y and , Ix z>,... progress on a more direct path toward the exact ground-state wave function than the modified-Lanczos/steepest-descent algorithms do. A strong indicator of the directness of that path is the sequence of angles (4.37)

k=1,2, ...

0,-------------------------, o n=14 o n=16 t:.

n=lB

-12+--------r--------r1---lI1-~-~----_l

o

10

20

30

40

k Figure 4-1: Logarithm of the relative deviation of the eigenvalue estimates Rk from the asymptotic value R~, plotted versus k (the number of iterations) for the 1D s= 1/2 Heisenberg antiferromagnet. The value of R~ has been approximated by our best estimate, which satisfies the convergence criterion (4.31) to within machine precision. The data points connected by solid lines have been obtained from the conjugate-gradient method and those connected by dashed lines from the steepest-descent method. Data for three different system sizes are shown.

Section 4-9

47

between the vectors IPk_l>' Ipy of successive iterations. In the steepest-descent/modified Lanczos method successive directions of the path toward the exact ground state are orthogonal to one another: = Alepo>, and lepo> is the ground-state wave function. Upon Laplace transform, 00

fdtei~ts(t) :: <Wol~I'I'O> o

::

~-H

<Wol'l'o>do(~)

(4.39)

we arrive at the relaxation function do(~)' the quantity investigated most directly by the recursion method. At the same time we recognize (4.39) to have the structure of a Green's function,

G(~)

::



=

~-H

-ido(~)

(4.40)

,

for the same state now expressed in terms of the normalized wave function luo> = 1'1'0>1112. Inserting 1 = L).,lep).,><ep).,1 yields the spectral representation of the Green's function:

G(~)

::

L ).,

<ep).,luo> :: ~ -E).,

1

L

<epolAAlepo> ).,

l<ep olAlep).,>1

2

.

(4.41)

~ -E).,

The spectral representation of the structure function is then a set of &-functions, obtained from the Green's function as follows: S(ro) :: -2<ep olAAlepo> limS[G(ro+ie)] :: 21tL l<ep olAlep).,>1 20(ro-E).,). £~o

).,

(4.42)

This representation has played an important role e.g. in the finite-size analysis of the T=O dynamics of quantum spin chains [Muller et al. 1981]. In those applications, expression (4.42) was evaluated from the complete set of eigenfunctions lep).,> and eigenvalues E)., determined by numerical diagonalization of H. For a system with a dynamically relevant subspace of finite dimensionality, the same set of 0functions (4.42) results via (4.12) from a finite continued fraction. In Sec. 4-4 we have presented the Lanczos algorithm as one of several iterative methods for the computation of the ground-state wave function of a model Hamiltonian. For any given normalized initial state luo>, the recurrence relations (4.17) produce an orthonormal basis of the invariant subspace in which luo> is located and a tridiagonal Hamiltonian matrix in that basis. For the eventual determination of the ground-state wave function the vector luo> can be chosen arbitrarily as long as it has a nonzero projection onto the ground state. For the particular initial state luo> = <wol'l'0>-1I21'1'O>, where 1'1'0> =Alepo> and lepo> is the ground state of H previously determined by the same method, the Lanczos algorithm produces a tridiagonal matrix (4.18) whose elements are at the

Section 4-9

49

same time the continued-fraction coefficients of the Green's function (4.40) or of the relaxation function (4.10). The connection between the tridiagonal Hamiltonian (4.18) and the continued-fraction representation of G(~) is readily established if we recognize that the resolvent (4.40) is in essence the OO-element of the inverse of the matrix previously determined by the Lanczos algorithm in tridiagonal form [Haydock 1980, Grosso and Pastori Parravicini 1985]: 1 - 1 = = [~l-H]~

G(~)

~-H

.

(4.43)

For a finite tridiagonal matrix fi, that matrix element can be expressed as the ratio of two determinants as follows:

G(~)

=

Tl(~)

.

(4.44)

To(~)

Here To(~) is the determinant of the full matrix ~l-fi, whereas Tl(~) has the first row and column omitted. Because of the tridiagonal structure of the matrix ~l-fi, these determinants can be evaluated by means of simple recurrence relations. Denoting by Vk(~) the determinant comprising the first k rows and columns of ~l-fi, we have (4.45a) (4.45b) (4.45c) and, for the general case, the recurrence relations

Vk+l(~) = (~-ak)Vk(~) -b;Vk_l(~)'

k =0,1,2, ... ,N

(4.46)

with Vl(~) =0 and Vo(~) = 1. Once the determinant To(~) = VM:~) of~l-fihas been composed in this way, these recurrence relations can be inverted and brought into the form

Tk(~)

_ Y_

-=-......,..",.,.. Tk+l(~)

~

ak

-b 2 Tk+2(~) k+l '

Tk+l(~)

k=0,1,2, ...

(4.47)

Inserting (4.47) into (4.44) recursively, yields the Green's function in the continuedfraction representation

50

Chapter 4

G(~) ~

b

-a _

o

~

Z 1

Z bz

(4.48)

-a 1 - - ; ; - - ~

-a z - ...

which is equivalent to the relaxation function (4.10). The original assumption that the matrix ~l-fj is finite can be abandoned without consequences other than that the infinite tridiagonal matrix yields an infinite continued fraction.

5 GENETIC CODE OF SPECTRAL DENSITIES It is tempting to invoke this term from biology to characterize the relationship between the recognizable information stored in the ~k-sequence and the properties of the associated spectral density. The analogy has some validity in three aspects: o The ~k-sequence is a code of retrievable information on specific properties of the spectral density (band structure, singularity structure, decay laws for (O~oo etc). o The ~k-sequence is generative in nature. It produces the relaxation function in the continued fraction representation, which, in turn, determines the spectral density. o The ~k-sequence leaves room for contingencies if convergence criteria are violated. The exact shape of the spectral density is then no longer determined uniquely by the continued-fraction coefficients. The main goal of this Chapter is to familiarize the reader with some of the patterns found in ~k-sequences that are readily translated into specific properties of the associated spectral densities. In subsequent applications we shall see that these properties are relevant (sometimes crucial) for the interpretation of the underlying physics. In addition to this primary goal, we wish to elucidate some interesting mathematical relationships between ~k-sequences and spectral densities, which are quite illuminating even if they have only little bearing on typical condensed matter applications.

5-1 Finite

~k-Sequences

The most elementary situation occurs if an application of the recursion method for the evaluation of a spectral density terminates spontaneously and thus produces a finite sequence, ~!' ... , ~K' Technically, the algorithm of the recursion comes to a natural stop when the basis generation procedure (3.19) yields a vector A+! of zero norm. This implies that ~K+! = 0, which terminates the continued-fraction representation (3.25) of the relaxation function at the K h level: co(z)

1

= --------~!

Z + --------

z

~

+ ------

z

+

(5.1)

~K

+-

z

Likewise, if the recurrence relations (4.5) produce the vector !fK+!> = 0, the implication is that bk+! = 0, which terminates the continued fraction (4.10). That function in turn has been shown (in Sec. 4-3) to be related to a function of the form (5.1).

52

Chapter 5

Expression (5.1) is a rational function Px:- .

(5.7)

Its L\k-sequence is known in closed form [Magnus 1985]:

~k(k+2P)

L\ - ~~n--=:-:--=-~ k -

(5.8)

(2k+2p -1)(2k+2p +1)

The first two terms of its asymptotic expansion read

_ 1 2 1-4p2 L\k - -C%[l + - - + ... ] . 4

4k 2

(5.9)

Note that the square of the exponent p, which specifies the singularity at the band edge in the spectral density, is determined solely by the amplitude of the leading liP correction in (5.9). Hence, sequences that converge from above toward L\. have p2~e (solid lines) and a case with ~o~o' then the spectral density has also a ~-function central peak. When we set ~e=~o' the gap goes to zero, the central peak disappears, and (5.20) reduces to (5.7) with ~=1/2. Now consider the function (5.20) representing a spectral density of that category. For any such function, the ~-1 and the ~ converge toward different asymptotic values, ~~o) and ~e), respectively. If the central peak is present (A > 0), then the ~k-l converge toward the lower

Section 5-4

asymptotic value (A~o) where

= AL )

and the ~k toward the higher value (A~e)

AL = {(Ofuax -Ofuin)2, An = {(Ofuax + Ofuin)2 ;

57

= An), (5.21)

otherwise (A=O) we have A~o) = AH and A~e) = AL" In either case, the rate of convergence depends on the function F(ro).

~k , . . . . - - - - - - - - - ,

~k , . . . . - - - - - - - - - - ,

~

1\

o

e

1\ o

1\

e

-Col

max

-Col

•

mm

o

k

CoI

min

Col

max

Figure 5·3: Prototype spectral density (5.18) with bounded support and a gap for the two cases !J.q>!J. e (solid lines) and !J.o
5-4 Spectral Densities with Unbounded Support In many-body dynamics, spectral densities with bounded support are realized only under exceptional circumstances, for example, when the relevant degrees of freedom are noninteracting or when selection rules screen out all but a very limited class of excitations. Under normal circumstances, spectral densities of many-body systems have unbounded support, Le. their spectral weight extends to ro~±oo. Such spectral densities are characterized by Ak-sequences which grow to infinity as k~oo. The growth rate of any such Ak-sequence is defined as the exponent A. which characterizes its average power-law growth,

58

Chapter 5

A.

k

(5.22)

kt.. ,

-

asymptotically for large k. In any given realization, the A.k may scatter considerably about the mean growth curve. These deviations determine the detailed shape of the spectral density, specifically its singularity structure. However, the growth rate alone determines the decay law of the spectral density,

$o(ro) = exp( -ofII..) ,

(5.23)

asymptotically for large ro [Magnus 1985, Lubinsky 1987]. In Chapter 7 we shall discuss the role of the growth rate of A.k-sequences for the identification of universality classes of dynamical behavior. The limiting decay law with growth rate A.=O corresponds to spectral densities with bounded support, which we have discussed in Secs. 5-2 and 5-3. A.k-sequences with linear growth rate (A.=I) are a common occurrence in quantum many-body dynamics, as we shall see in the applications discussed later on. This type of behavior is exhibited by the following model spectral density:

$ o(ro)

=

IX

21t ro e -oh~ . Cl\!r( a + I) Cl\!

(5.24)

2

The corresponding model fluctuation function is a degenerate hypergeometric function, eo(t)

1 1 2 2 = $ (a+l - ; - ; --%1 ) .

2

2

(5.25)

4

The A.k-sequence for this model spectral density is known in closed form: ~-l =

I 2

'2Cl\!(2k-1 +a), A.2k =

I 2

'2 CJt(2k)

.

(5.26)

In this simple pattern, the (average) linear growth of A.k determines the Gaussian decay of $o(ro) at large ro, the slope of the line ~ determines the frequency unit COo in (5.24), and the vertical displacement of the ~-l from that line determines the exponent a of the infrared singularity (see Fig. 5-4 for two specific cases of that model spectral density and its A.k-sequence). Growth rates 1..>1 are not uncommon in many-body dynamics, especially in classical systems with nonlinear dynamics. Consider the model spectral density,

21t IA.c%

ro

IX

' $o(ro) = --,.----1 exp( -lro!Cl\!l v ",) , n A(l +a)] Cl\!

(5.27)

2

of which (5.24) is a special case. We know the frequency moments of (5.27) in closed form,

Section 5-4

= ~n A(l +a +2k)]tr[ A(1 +a)]

M 2k

2

2

59

(5.28)

,

but not the ak-sequence. However, the latter can be generated numerically from moments by means of the transformation formula (3.33).

fo k . . - - - - - - - - - _ _ , ,

a=-1/2 k

k

;'

--

,'"

...

;';'

/

/

/

/

;'

... - ....,

\

I

\

\

\

\

I

\ \

;'

\

,

//

I

I

/

;'

...-....,,

\

I

\

I

\

\

\ "

o

"

...........

_-

"'0

CA)

Figure 5-4: Model spectral density (5.24) with unbounded support and Gaussian decay at high frequencies for the two cases a=-1/2 (solid lines) and a=2 (dashed lines). The Aksequences (5.26) for the two cases are displayed in the insets [adapted from Viswanath et al. 1994].

For the special case A.=2 of (5.27), which corresponds to a quadratic growth rate of the ak-sequence and an exponentially decaying spectral density, we obtain the following expression for the fluctuation function: cos[(1 +a)arctan(Cl\>t)]

Co(t)

= ----=----(1 +

ott

(5.29)

2)(0. ...1)/2

For <X=O, this reduces to a Lorentzian, whose spectral density is a simple exponential: ,cI»o(ro) = _l_e -lro/C%' •

1

1

+ott

2

Cl\>

(5.30)

60

Chapter 5

However, the simplest pattern with quadratic growth rate, the uniformly quadratic dFsequence, (5.31)

belongs to the functions Cl> o( (0)

7t 7t00 = _sech(_),

Cl\!

2Cl\!

= sech(Cl\!t)

Co(t)

.

(5.32)

For fluctuation functions CoU) which are entire functions of (complex) time the growth rate A. of the d k-sequence is expressible as

A. = 2(p -1)/p

(5.33)

in terms of the growth order p. The latter quantity specifies the growth of Co(t) for large imaginary times [Roldan, McCoy, and Perk 1986]: Co(it) - exp(t p )

•

(5.34)

A proof that Co(t) is entire exists only for fairly special circumstances [Araki 1969]. Note that the growth rate A.=2 represents the limiting case of infinite growth order (p=oo). dk-sequences with growth rates Q2 do not represent fluctuation functions which are entire functions of t. One case in point is the function Co(t) in (5.32), for example. It is a realization of A.=2 and evidently not entire. For Q2, the continued fraction representation (3.25) does no longer converge; the moment problem does not have a unique solution; the spectral density cannot be reconstructed uniquely from the d k-sequence. The difference between spectral densities with the same d ksequence is, in general, an oscillating function.

5·5 Spectral Densities with Unbounded Support and a Gap The presence or absence of a gap in a given spectral 'density of a many-body system may serve as an important indicator of a specific type of ordering in the system and provide crucial clues on the nature of the excitations (quasi-particles) which govern the dynamics of that system. In Sec. 5-3 we have already learned how to recognize the presence of a gap in spectral densities with bounded support and how to determine its size by analyzing the dk-sequence produced by the recursion method. How do we recognize by the same methods the presence of a gap in spectral densities with unbounded support? What are the characteristic patterns in the associated dk-sequences that indicate the presence of a gap, and what are those that indicate the presence of a 0(00) central peak in the gap? Consider the model spectral density cI>0(00)

= 27tAB(00) + 2{i (l -A)e(lool-O)e -(ICOI-n>2tot. '%

(5.35)

It has unbounded support and a gap of width 20 centered at 00=0. For A=O and 0:0, expression (5.35) reduces to a pure Gaussian, whose dk-sequence grows linearly

Section 5-5

61

with k, !:ik = roo2k12, as discussed in Sec. 5-4. For the investigation of the effect of a gap and the effect of a central peak in the gap on that !:ik-sequence, we have detennined the moments of (5.35) in closed fonn [Viswanath et al. 1994], k (

M 2k = 21t(1-A)L

m=O

)

2md-(k-m)

2k Ob 2m 2m

(2m-l)!! (5.36)

+ 2{i(1-A)E ( 2k l02(k-m)-lroo2m +1m !, k=I,2, ... m.() 2m+l

J-

from which the !:ik-sequence can be derived numerically by means of the transfonnation fonnula (3.33). The results for two cases are plotted in Fig. 5-5. The effect of the gap is to split the !:ik-sequence into two subsequences !:i2k and !:i2k_l that still grow (roughly) linearly, but with different slopes. In the absence of a central peak, the !:i2k_l grow more steeply than the !:i2k (solid lines, left inset). If the ~ grow more steeply, this is an indicator that a central peak is present (dashed lines, right inset).

1\ ,----------::>1

k

k

-0

o

o

c.J

Figure 5-5: Model spectral density (5.35) with unbounded support and a gap for the two cases A=O, roo=2!l (solid lines) and A=1/2, roo=20 (dashed lines). The 4 k-sequences for the two cases are displayed in the insets [adapted from Viswanath et al. 1994].

62

Chapter 5

The same effect of a gap and a central peak within the gap can be observed even more clearly in a class of model spectral densities with growth rate 1..=2. Consider the two Jacobi elliptic functions .... (c)

q.oo

_

(t) - cn(C%t,K),

.... (d)

q.oo

_

(t) - dn(C%t,K) ,

(5.37)

which we assume to play the role of (normalized) fluctuation functions of some hypothetical dynamical system. For Kbc )(oo) and 4>bd)(oo) consist of infinite sets of ~functions at equidistant frequencies. For large 00, the spectral weight of these individual lines approaches zero exponentially, in agreement with (5.23) for 1..=2. The discreteness of the spectrum implies an energy ga~. Consider first the function 4>bc (00). The size of the gap is 41t divided by the period of cn(root,K). There is no central peak. The ~k-sequence for that spectral density is known in closed form:

141-1

= olo(2k-1)2,

141

= olor(2k)2 .

(5.38)

In a plot ~1c) versus 0 we have again two lines with different slopes similar to the example with growth rate 1..=1 depicted in Fig. 5-5 (inset left). The main difference of the spectral density 4>bd)(oo) with respect to 4>bc )(oo) is that the former has one additional line at ro=O - the central peak in the gap. The consequences for the ~k-sequence are very similar to what we have already observed in the context of Fig. 5-5:

~-1

= olor(2k-l)2,

~

= olo(2k)2 .

(5.39)

tJ4P

It is now the subsequence which takes off more steeply in a plot ~1d) vs 0. All combined, the analysis of the ~k-sequence as obtained by the recursion method may enable us to make a reliable prediction about the existence of a gap and the presence of a central peak within the gap.

5-6 Orthogonal Polynomials It is appropriate to conclude this Chapter, which is all about the relationship between patterns in the sequence of continued-fraction coefficients and properties of the spectral densities, with yet another such relationship. For that discussion we use the concepts developed in Chapter 4, specifically the double sequence of numbers {ak> b~}. They play the role of continued-fraction coefficients in the relaxation function (4.10) or the role of matrix elements in the tridiagonal Hamiltonian (4.18) as produced by the Lanczos algorithm. We recall that for a given Hamiltonian H and a given (normalized) initial state 1"0>, the {ak> b~}-sequence of the (normalized) structure function 5 0(00) in whose properties we are interested, is determined iteratively by means of the recurrence relations (4.17). The same set of recurrence relations can now be turned around and be used for given {ak' b~} to generate a complete set of orthogonal polynomials Pk(oo),

Section 5-6

1

Pk+1(ro) = -[(ro-ak)Pk(ro) -b~k-l(ro)], k=0,1,2, ...

bk+1

63

(5.40)

with initial conditions P_1(ro)=0, Po(ro)=1 [Grosso and Pastori Parravicini 1985]. The structure function So(ro) associated with the double sequence {ak' b~} plays the role of a weight function in both the orthogonality and completeness conditions: +00

(5.41)

So(ro)L Pn(ro)Pn(cJ)

= 21tO(ro-cJ)

.

(5.42)

n

Table 5-1 lists some of the better known sets of orthogonal polynomials along with the associated structure function and continued-fraction coefficients. Orthogonal Polynomials

Frequency Interval

Tchebycheff 1sI kind: Tn(ro)

[-1, 1]

Tchebycheff 2nd kind: Un(ro)

[-1, 1]

Legendre: Pn(ro)

[-1, 1]

Laguerre: Ln(ro)

[0, 00]

Hermite: Hn(ro)

[-00, 00]

Structure Function

2

~ 4~ 1t fXl -loo)

Continued-Fraction Coefficients ak = 0

b~ = (l +Ok,I)/4 ak = 0

b~ = 1/4 ak = 0

b~ = ~/(4~-1) 21t e -Q)

ak = 2k+l

b~ =~ {41te;d

ak = 0

b~ = k/2

Table 5-1: Selected sets of orthonormal polynomials and associated weight functions (structure functions 80(00)) generated by the recurrence relations (5.40) from specific double sequences {ai. The recurrence relations (4.5) terminate spontaneoulsy during the first step, .

!fo> = qlO> =

J

1

2mroo

11>,

ii = H -Eo = rooa t a

,

(6.5a)

Section 6-}

65

(6.5b)

aO

2

(6.5c)

= '%' If}> = 0, hI = 0 ,

and yields the relaxation function (6.6) from which we infer the structure function S(ro)

= lim 29t[dO(ro+ie)] = ~5(ro-,%) m,%

E~O

.

(6.7)

Path #2: For the calculation of the same structure function of the quantum harmonic oscillator at arbitrary temperature, we employ the Liouvillian representation of the recursion method. The most direct path toward that goal uses the inner product (3.9). The first iteration of the recurrence relations (3.19) is carried out as follows:

10

'=

(6.8a)

q,

(fo,/o) = {») reduces to that of the harmonic oscillator. The two-spin correlation function <Sl(t)Si+1> is then governed by linear spin-wave excitations alone. The determination of the dynamic structure factor

Sxx(q,ro)

=E

e iqn

n

......

f

(6.23)

dteirot<st(t)SI:n>

in the ferromagnetic ground state, l«!lo> = IF> = liit....i>, with all spins aligned in the positive z-direction is then just another simple exercise. The ground-state energy is Eo = -N/4. In the Hamiltonian representation of the recursion method, the following steps lead to the desired result:

!fo> = SqXI F>, Sx= q

l~iqlsx

(6.24a)

-LJ e l '

1N l

1

(6.24b)

= '4' ao = roq = l(1-cosq) ,

(6.24c)

Sxx(q,~

= 21t 0 (CO-roq)

(6.24d)

.

In the Liouvillian representation, different steps lead to the same result:

(6.25a)

= r;;;LJ 1 ~eiql{SZSY (e iq -l) f 1 = I'[HSx] 'q I 1"'1 VN I

+

SYSZ (l-e iq )} I 1...1

(625b)

,.

(6.25c)

Section 6-3

«I».u(q,ro)

69

= 2(fo,fo) lim 9t[co(£ -iro)] £~o

=

(6.25d)

~ {O(ro-roq)+O(ro+roq)} , 7t

(6.25e)

S.u(q,ro) = ZO(ro-roq ) .

This exact solution is readily generalized to spin quantum numbers s> 112 and to higher-dimensional lattices. Now suppose we wish to determine the structure function S.u(q,ro) or the spectral density «I».u(q,ro) for the Heisenberg antiferromagnet (J is known to decay algebraically as -(-1)nln for large distances [Luther and Peschel 1975]. o No spontaneous termination takes place in the application of the recursion method, implying that the structure function S.u(q,ro) for fixed wave number now has a continuous spectrum. Any serious attempt to deal with these complications would employ one of the algorithms presented in Secs. 4-4 to 4-7 for the determination of the true (finitesize) ground-state wave function and then proceed in the Hamiltonian or the Liouvillian representation of the recursion method. In Sec. 4-8 we have already reported how to execute the first part of that calculation for the model system at hand. How best to carry out the remaining tasks will be discussed in Chapter 11.

6-3 Lattice Fermions Consider the ID s=1I2 XXZ model, specified by the Hamiltonian N

H

= -E [J(StSI:1

+

1=1

S/SI~I)

+

(6.26)

J.J/SI~d .

It is well known that this model is equivalent to a system of spinless lattice fermions [Lieb, Schultz, and Mattis 1961; Katsura 1962] H

= E £o(k)a1a k k

+

E

_1 V(q)p(q)p(-q) , 2N q

£o(k) = -Jcosk, V(q) = -2Jzcosq , p(q) =

E ala

(6.27a)

k+q'

(6.27b)

k

The mapping between these two representations of the same system is provided by the Jordan-Wigner transformation,

70

Chapter 6

t [.11tL-Ja ~ ta ] S1+ == S1x +1·S1y .. a,exp j j

,

j~ .

(6.30)

k'

The equations of motion for the two functions G+ and G-;- read:

~E for special cases of the classical XYZ -dimer, H

.c;' x x = -J~l S2

y y z z - Jl1 S2 - J~l S2 .

(6.52)

The simplest case pertains to parameter values Jx=Jy=O, Jz=1 (X dimer). For the apprentice's convenience, we have reproduced in Table 6-1 the steps of the first three iterations. The pattern of this sequence is very simple and readily recognized: 2

!:!. _ k -

2

k (2k -1)(2k + 1) ,

(6.53)

It is the special case ~=O, 0l0=1, of the model sequence (5.8). For these parameter values, the model spectral density (5.7) reduces to the function

2A different application of the dk-sequence (6.53) was reported by Lee and Hong [1984] in the 3D Sawada model.

Section 6-5

et> o((0) = 1t eo -lroI)

75

(6.54)

.

The associated spin autocorrelation function <Sj(t)Sj> is then a spherical Bessel function:

= sint .

eo(t)

(6.55)

t

Our second example is the XX dimer, characterized by the parameter values (1x=Jy= I, Jz=O). The dynamics of that model belongs to an entirely different

category. This is reflected, for example, in the L\k-sequence of the same spin autocorrelation fucntion <Sj(t)Sj>. It does not converge to a finite value, but instead grows to infinity quadratically (shown in Fig. 7-4 below). That dramatic change in pattern is attributable, as we shall see, to the nonlinear nature of the underlying dynamics. An extensive discussion of the different categories of dynamical behavior - we call them universality classes - will be given in Chapter 7. In Chapter 9 we shall discuss a third case of (6.52), the XXX dimer (1x=Jy=Jz)' for the demonstration of yet another feature of dynamical behavior.

10

=

st

I1

=

iLlo = -S/S2

~ (/1 ,/1) = «SIY)2(S2z)~ =

12

=

zLII +L\I!0

.

x

1

x2

~ (/0'/0) = «SI) > = -

3

z

z2

1

"9

~ L\I =

1

"3

1 x 3

= -SI (S2) +_SI

z ~ - -«SI) 2 X2(S2) z ~ + -«SI 1 )~ = ~ (/2 '/2) = «SI )2(S2)

3

4 135

9

4 15

~~=-

13

=

. zLI 2 +L\~I

y

S z3

y

2

9 15

= SI ( 2) --SI

z6

z 2

ys

18 15

y

2S z4

81 225

~ (/3'/3) = «SI) (S2) >-_«SI) ( 2) >+-«

S IY)2(S 2z)~ = - 4

525

9 ~ L\3 = 35 Table 6-1: The first three iterations of the recurrence relations (3.19) for the classical X dimer, specified by the energy function (6.52) with J;FJrO, Jz=1.

7 UNIVERSALITY CLASSES OF DYNAMICAL BEHAVIOR In most studies of dynamic correlation functions the focus is on their long-time asymptotic behavior and on the singularity structure of the associated spectral densities. These properties reveal important information on the nature of the physical processes which govern the dynamics of the system under given circumstances. However, the analysis of the same dynamic correlation functions from a quite different perspective can be equally useful and revealing. The focus there is on the properties of spectral densities at high frequencies, specifically their decay law, expressible as in (5.23), in terms of a characteristic exponent A.. The information contained in A. about the underlying dynamical processes is in some sense complementary to that inferred from the singularity structure of spectral densities. Since the characteristic exponent A. is equal to the growth rate of the ~k­ sequence for the spectral density as defined in (5.22), the recursion method is the ideal calculational technique for such dynamical studies. However, we have yet to unlock the dynamical information contained in the characteristic exponent A. for general situations. In an effort to gain an intuitive understanding for the connection between decay laws of spectral densities and other dynamical properties, we report here a study which employs this type of analysis for a particular, exactly solvable model: the equivalent-neighbor XYZ model [Liu and Muller 1990]. Its dynamical properties depend strongly on the symmetry of the exchange interaction, and the analysis of dynamic correlation functions can be carried out to a considerable extent. The results of that study can be used as the basis for a classification of dynamical behavior in terms of the characteristic exponent A.. 7-1 Dynamics of the Equivalent-Neighbor XYZ Model Consider an array of N spins interacting via some model-specific spin-pair coupling of uniform strength r. In order to ensure that the free energy is extensive, the coupling strength must be scaled like r = llN. In this scaling regime, the equivalent-neighbor spin model is a microscopic realization of mean-field theory, but it has no longer any intrinsic dynamics. The right-hand side of Hamilton's equation for individual classical spins, dS/dt = -SI x aHlaSI, vanishes in the limit

N~oo.

A nontrivial intrinsic dynamics (for N~oo) can be restored, at least in the paramagnetic phase, if the spin coupling is scaled differently: r = l1N1I2. Timedependent correlation functions at T=oo are then meaningful and interesting quantities. They are the object of investigation in what follows. The two scaling regimes are best understood by noting that the thermodynamic properties of the equivalent-neighbor spin model are governed by the mean value of the magnetization vector (which is the basis of Landau theory), whereas the dynamical properties are determined by the fluctuations about the mean value.

77

Section 7-1

The Hamiltonian of the classical equivalent-neighbor XYZ model reads N

E

H = __1_ 2.jN jj=l.j#j

[J~tS/

+ J,f/S/ + JzS/S/J .

The equations of motion for the classical spin variables a j

dS _ J &

4.0

2.0

o.0 --t-~X(co)sof tile classical equivalent-neighbor ><XZ model at T=oo. The curves represent the exact result (7.9a) for six different values of uniaxial anisotropy, here parametrized as J = sin(ng), Jz = cos(ng). The inset shows the same function for parameter values g = 0.29, 0.28, 0.27, 0.26, approaching the XXX model (g = 0.25), for which case (7.9a) reduces to (7.9b) [from Liu and Muller 1990].

Note that the long-time aymptotic behavior of these correlation functions depends on the amount of uniaxial exchange anisotropy, whereas the decay law of the spectral densities at high frequencies is always Gaussian, independent of the model parameters. The Gaussian decay of spectral densities represents one of four different universality classes of dynamical behavior that are realized in the context of the general equivalent-neighbor XYZ model. We shall see that they can be interpreted in terms of basic notions of classical dynamics. Before we proceed with a discussion of these universality classes (in Sec. 7-5), we must adapt the recursion method to the special requirements of equivalent-neighbor models. 7-3 Recursion Method Applied to Equivalent-Neighbor Spin Models At first glance, it seems that the infinite-range interaction in equivalent-neighbor spin models makes them inaccessible to any useful analysis by the recursion method except for small N: The opposite is true. The metamorphosis as N~oo of the equivalent-neighbor XYZ model into a physical ensemble of two-body systems can

Section 7-3

81

be exploited to boost the computational efficiency of the recursion method substantially. We start out with the formulation of the method for classical spins as outlined in Sec. 6-5 and rewrite the Hamiltonian (7.1) in the form I ~ I ~ ~ a2 H = - - - L.J JaMaMa + - - L.J L.J Ja(Sj) ,

2{N a=XYZ

2{N a=xyz j=l

(7.11)

where N

Ma

= Lst,

(7.12)

a.=xyz.

j=l

The Poisson brackets for the two sets of variables {stJ.,S!} = fJjjLEa[3ySl,

Sf and Ma are

{Sja,M[3} = LEa[3ySl '

y

y

(7.13)

{Ma ,M[3} = LE a [3yMy . y

In the recursion method as applied to the spin autocorrelation functions <Sf(t)Sf>, all inner products to be evaluated have the following general structure: (7.14)

Hence all terms except the leading (N-independent) one represent finite-size corrections. It turns out that only the Ma-terms in (7.11) contribute to the dynamics of the infinite system. Any surviving contribution to the vector A in the orthogonal expansion (3.17) has the general form N-(mx+my+mz)l2st(Mx)mX(MytY(Mz)mZ .

(7.15)

All nonvanishing inner products, expanded in inverse powers of N, factorize to leading order: «Sja)2

IT

y=xyz

N-m., My2.m.y>

=

«Sja)~ IT N-mY <My2.m.y>[1 +O(N-1)]

(7.16)

y=xyz

with (7.17) Exactly the same inner products are obtained from a physical ensemble of the 2-degrees-of-freedom system consisting of a spin Sj of unit length coupled parasitically (Le. with no dynamical feedback) to an autonomous I-spin system, a spin cr of unit rms length driven by the Hamiltonian (7.18)

82

Chapter 7

This representation is particularly suitable for evaluating !:ik-sequences of the infinite-N equivalent-neighbor XYZ model by means of the recursion method.

7-4 Quantum Equivalent-Neighbor XYZ Model Through minor modifications, the recursion method can be adapted to the dynamics of the quantum equivalent-neighbor XYZ model, specified by Hamiltonian (7.1) or (7.11) with the Si now representing spin-s operators. We employ the quantum Liouville equation (3.1) instead of its classical counterpart (3.11), replace the symplectic structure (6.49) by the commutator algebra for quantum spins, [Sia,S!] = iOi/E£apySl '

(7.19)

y

and use (for T=oo) the inner product (A,B) = Tr(AB). In spite of the structural similarity of the elements which go into the classical and quantum versions of the recursion method, the resulting dynamics is, in general, quite different for the two cases. However, the equivalent-neighbor XYZ model is atypical in this respect. All inner products have the general structure (7.14) in both the quantum and the classical cases, and the dynamics of the infinite system is determined by the leading term alone. All terms in I k which contribute to leading order in (7.14) contain commuting operators only - operators pertaining to different sites of the array. The net result is that the coefficients !:ik of <S:'(t)S? for the infinite quantum equivalent-neighbor XYZ model differ from those of its classical counterpart only by a multiplicative conStant (which depends on s), amounting to a different time scale in the dynamical correlation functions. This explains the observation that the results of Liu and Muller [1990] for dynamic correlation functions of the classical equivalent-neighbor XXZ model reported in Sec. 7-2 are fully consistent with the results previously obtained by Lee, Kim and Dekeyser [1984] for the quantum spin1/2 counterpart of that model (see also Dekeyser and Lee [1991]).

7-5 Prototype Universality Classes In integrable classical dynamical systems, the growth rate A. of the !:ik-sequences as defined in (5.22) for specific autocorrelation functions is basically dominated by two factors: Factor A depends on whether the equations of motion are linear or nonlinear. Each harmonic mode contributes exactly one o-function to the spectral density (at 00>0), whereas each anharmonic mode contributes an infinite set of &-functions, at frequencies with no upper bound. In nonlinear systems, factor A is governed by the large-m decay law of the line intensities for single modes. Factor B is governed by the distribution of fundamental frequencies pertaining to individual linear or nonlinear modes. That distribution depends sensitively on whether the size of the system is finite or infinite and (for infinite systems) on whether the interaction range is finite or infinite.

o

o

Section 7-5

83

The effect of each factor on the large-ro decay law of the spectral density is expressible in tenns of a distribution function: factor A by the spectral-weight distribution Pn(n) of individual modes and factor B by the distribution Po.(O) of fundamental frequencies of these modes. The large-ro decay law of the spectral density is then obtained from these distributions by the following construction: 00

(7.20)

(ro-nO) .

o

0

For distributions with asymptotic decay laws of the fonn P n(n) - exp( -n Cl) , Po. - exp( -03)

,

(7.21)

the large-ro decay law of the resulting spectral density as obtained from (7.20) is given by

(n-1) l>(n-l)

-e -n -e -n

size of system finite infinite finite infinite

A

Po.(O)

o(co) and growth rate A. of the associated Ak-sequence for the four different universality classes of dynamical behavior realized in the classical equivalent-neighbor XYZ model. The four classes arise as the product of the two factors A (type of dynamics) and B (size of system). each represented by a distribution for which there are two distinct realizations [adapted from Liu and Muller 1990].

84

o

Chapter 7

Bounded support (A=O): Quite generally, .:lk-sequences with zero growth rate are realized in linear dynamical systems with a finite number of degrees of freedom. The finite-N XXX case is such a system. The .:lk-sequence for the case N=2 is plotted in Fig. 7-2. It converges toward a finite asymptotic value .:loo in an alternating approach. This situation is represented by the first row of Table 7-1 and by the limiting case (a.,~) (00,00) of (7.22). The exact spectral density will be reconstructed in Chapter 9 from the .:lk-sequence.

=

2.0

2-spin XXX model a a

<So (t)S. > 1

1

1.5

1.0

0.5

o. 0 +-__r_--.---.----"T---,.-r--.,.---r-~_r___r___r__r____r---,....__.,.__-r--._-l 18 o 2 6 8 10 12 14 16 4 k Figure 7-2: Sequence of continued-fraction coefficients !:J.k versus kfor the spin autocorrelation function at various values of J, Jz' all of which show indeed linear growth rate. This situation is represented by the second row of Table 7-1 and described by (7.22) with (a.,~) (00,2). Only the sequence for J=O is purely linear, .:lk=kJ/13, representing expression (7.10). Throughout the regime O<J<Jz the.:lk oscillate about the

=

Section 7-5

85

line kJ//3 (shown dashed in Fig. 7-3). These oscillations persist as k~oo if 1j2<J<Jz and thus determine the singularity structure of the spectral density (7.9a) and the power-law long-time tail of the correlation function (7.8a). For O<J<Jj2, on the other hand, the oscillations damp out as k~oo, which is illustrated for two cases in the inset to Fig. 7-3. These ak-sequences describe spectral densities with no power-law singularities and correlation functions with no power-law long-time tail. 0.05

40.0 t')

"..lI:

I ..lI: at T=oo of the classical two-spin XX model, (6.52) with Jr Jy=:1. J;t=0. which is equivalent to (7.1) for N=2 and Jr J-2 112 , J;t=0. The quadratic growth rate (A.=2) for this function is demonstrated by the excelfe"nt fit of the regression line [from Liu and MOller 1990j.

o

Stretched exponential decay (A.=3): For realizations of this last universality class, consider the collective-spin autocorrelation function of the XYZ case with N=oo, specifically the function for lx=(l +y)/2, ly=(l-y)/2, lz=O and four different values of the parameter y. For the two cases with uniaxial symmetry, y=O and "(=1, the dynamics is linear, whereas for the two intermediate cases with biaxial symmetry, "(=1/4 and "(=3/4, it is nonlinear and describable in terms of Jacobi elliptic functions. In all of these cases, the distribution of fundamental frequencies of individual (linear and nonlinear) modes is characterized by a Gaussian decay (factor B), but for the nonlinear cases, the large-m behavior of the spectral density is further governed by the exponential decay of line intensities at multiples of the

Section 7-6

87

fundamental frequencies (factor A). These situations are represented, for the linear and nonlinear cases, respectively, by the second and fourth rows in Table 7-1 and by (7.22) with exponent values (a,~) equal to (00,2) and (1,2). Figure 7-5 shows for the four specified cases !:1k versus k in a log-log plot as determined computationally by the recursion method. The switch from linear to nonlinear dynamics causes the growth rate to jump from 1..= I to

1..=3.

1000.0

• J x =0.B75 J y =0.125 • Jx =0.625 J y =0.375

100.0

10.0

o J x =1.0 Jy=o. o J x =0.5 J y =0.5

1.0

0.1 +-----.------,--..----,---,r--r----.--.-.-----..-----r-----; 40 6 8 10 20 2 4 1

k Figure 7-5: Log-log plot of sequences l:1 k versus k for the collective-spin autocorrelation function at T=oo of the 10 classical and quantum spin-1/2 XX model (7.33) with .1=1. The sequence for the quantum model is exactly known, li.k =k/2. The (solid) regression line determined for li. 1 ,••. ,li.7 of the classical model has slope 1..=2.19. Lines with slope 1..=1 and slope 1..=2 are shown dashed [from Liu and Muller 1990].

(vii) Time-dependent correlation functions with singularities are not unheard of in

otherwise well-behaved classical many-body systems with Hamiltonian dynamics. For the case of the completely integrable logarithmic Heisenberg model [Ishimori 1982; Haldane 1982; Papanicolaou 1987], N

H =

-L In(l +S i'Si+l) , i=l

(7.34)

92

Chapter 7

such singularities even make it to the real t-axis, at least for finite N. The exact T=oo autocorrelation function for N=2 reads: Co(t)

= -!.. 2

+

-!..(1 +-!"t 2)cost - t(~ +_I_t 2)sint 2

6

6

12

+ t 2(1 +_1_t 2 )ci(t)

12

= t 2(1 + _1_ t 2)lnt + regular terms .

12

This function is no longer describable in terms of .1k-sequences.

(7.35)

8 TERMINATION OF CONTINUED FRACTIONS: ATTEMPTS AT DAMAGE CONTROL In almost all practical applications of the recursion method, we are confronted with a problem that cannot be ignored. The implementation of one or the other version of this calculational technique enables us to determine only a limited number of continued-fraction coefficients ~1""~K before we run out of computational power (CPU time or memory space). This produces an incomplete continued fraction (3.25) for the relaxation function co(z). How much worth is an incomplete continued fraction? We might say, it is as much worth as an (incomplete) K-term asymptotic expansion (3.32) of the same function co(z) or a K-term short-time expansion (3.29) of the (normalized) fluctuation function Co(t). However, we must not ignore the following difference: An incomplete short-time expansion is a well-defined function. It represents an approximation to the function Co(t) within a certain radius of convergence. An incomplete continued fraction, by contrast, is in itself a meaningless object, somewhat like a fraction with the numerator or the denominator missing. The use of incomplete continued fractions as they invariably emerge from nontrivial applications of the recursion method depends on an acceptable scheme for the artificial completion of these nonentities. The objectives underlying this process must be the following: (i) Terminate the incomplete continued fraction artificially to make it a meaningful mathematical object - a function co(z). (ii) Use a termination scheme that does not violate the general constraints imposed on the relaxation function co(z) (causality, analyticity etc). (iii) Select a termination scheme which transforms the incomplete continued fraction into the best possible approximation to the relaxation function co(z) based on the maximum amount of information that can be extracted from the sequence of numbers ~1""'~K' Achieving the third goal looks like a formidable challenge. If an infinite ~k­ sequence is necessary to fully determine the function co(z), then knowledge of the first K coefficients ~K does not add up to any measurable amount of information in the worst possible scenario, and the recursion method would then be hopelessly inadequate. The power of the recursion method derives from the very peculiar way in which the information contained in the ~k-sequence is translated into properties of the associated spectral density. In the following, we discuss a number of different strategies which have been used to achieve the goals (i)-(iii) stated above. They range from acts of sheer desperation to more or less sophisticated attempts at controlling and minimizing the damage done by artificially terminating a continued fraction. In Chapter 9 we shall propose a particular scheme of implementing the recursion method specifically designed for the reconstruction of spectral densities from incomplete continued fractions.

94

Chapter 8

8-1 Cut-OtT Termination The crudest way of tenninating an incomplete continued fraction is by means of setting the first unknown coefficient equal to zero, aK+1 O. The relaxation function is then a finite continued fraction

=

co(z)

1

= ---~-­

a1

z+----:--~ z+----z+...

(8.1)

a

... +...!.. z

with K +1 poles on the imaginary axis. It yields a multiply periodic fluctuation function Co(t) of the form (5.3) and a spectral density 4>0(00) of the form (5.2), which consists of (K+l)12 pairs of ~functions. For a dynamic correlation function which in reality has a continuous spectrum this is worse than a caricature. The cut-off termination has many advocates among the users of the recursion method. Their argument in justification of its use may be paraphrased as follows (e.g. for the case of an interacting quantum spin system): Finite systems generate only a finite number of frequencies in the time evolution. The relaxation function of any such system can be rigorously expressed in terms of a finite continued fraction. Hence the error caused by the cut-off termination cannot be more serious than that of a typical finite-size effect. Its impact weakens as the number of coefficients used in (8.1) increases. Even relatively small systems yield hundreds if not thousands of poles along the imaginary z-axis. If the spectral density is evaluated, via (3.28), but with E kept nonzero, the hundreds or thousands of &-functions are spread into Lorentzians and Itlimic a continuous spectrum. For the right choice of E, these Lorentzians overlap sufficiently to bring forth a spectral-weight distribution that reflects some of the major features of the function which it is intended to approximate.

Unfortunately, it must be said that the argument is flawed, and the analysis of results based on this approach runs a considerable risk of mlsinterpretation. The problems with the cut-off tennination are best understood after we have identified and distinguished the following specifications: N number of particles or degrees of freedom (system size); L number of frequencies in the time evolution (dimensionality of the dynamically relevant Hilbert subspace of the Liouvillian); K (=2L-l) number of ak in the finite continued fraction (8.1); J number of k which are unaffected or only little affected by the finite size of the system. Our observation for generic quantum spin systems is that L and K increase exponentially in N, while J increases proportional to some power of N. In a system with nearest-neighbor interaction on a d-dimensionallattice, we have 21+1 .. N Ud. For an illustration consider the ID s=1I2 XX model (7.33) with periodic boundary conditions. We have computed ak-sequences of the T=co spin autocorrelation function <S[(t)S[>, for different system sizes N. We find that the number K(N) of nonzero continued-fraction coefficients does indeed increase very rapidly with

a

N:

95

Section 8-1

K(2)

= 1,

= 8,

K(4)

K(6)

= 36,

K(8) > 80 .

(8.2)

For given N, the !:J..k's are unaffected by the finite size of the system for k up to l (8.3)

J(N) = N-1 .

Beyond that mark, the !:J..k's start to deviate at first gradually and then erratically from the sequence (6.46), !:J..k=kJ212, which pertains to the infinite system. This is illustrated in Fig. 8-1. 8 10

1.1 N=oo

8

6

~

1 .

(8.32)

Figure 8-7 shows the m-dependence of the extrapolated coefficients !!J.K +1 for the case where the Gaussian-memory-function approximation is implemented at K+l=4 (short-dashed line). What strikes the eye is a serious mismatch in slope between the linear growth of the new coefficients extrapolated according to the recipe (8.32) and the slope of the average linear growth of the exact coefficient. This mismatch becomes worse for larger K, where the approximation is supposed to be better. The trick which is justifiable in the case of the stationary-memory-function approximation cannot be transcribed without major modifications to Gaussian-type memory functions. The Gaussian-memory-function approximation is manifestly not convergent, not even for a pure Gaussian. Linear-extrapolation approximation [Morita 1975]: An alternative way to extrapolate a !!J.k-sequence with linear growth has been employed in a number of applications. It also uses a Gaussian-type memory function, but one with two parameters, given by the last two explicitly used exact coefficients, !!J.K, !!J.K+1• The

108

Chapter 8

prescription for extrapolating the ~k-sequence which is equivalent to such a twoparameter memory function reads: (8.33) This extrapolation procedure is implemented at level K=5 in the ~k-sequence shown in Fig. 8-7 (long-dashed lines). It is clear that the linear-extrapolation approximation is beset by essentially the same problems as the previous scheme. For a pure Gaussian, the extrapolation (8.33) is exact, but that is the exception. Whenever we have (~K+r ~K) < 0, which occurs not infrequently, (8.33) loses all legitimacy. It would be preferable to use an approximation whose effective extrapolation of the ~k-sequence follows the linear regression line of the explicitly known coefficients (shown solid in Fig. 8-7). We shall return to this suggestion in Chapter 9. The termination procedures discussed in this Chapter seem to operate under the erroneous assumption that in approximating x-l,

(9.24)

the model ~k-sequence (5.8), f! k

olcJc(k +2~)

= ....",....,,....-,,..,.....--:":"~----,...,,-'"""""'

(9.25)

(2k +2~ -1)(2k +2~ + 1)

and the model relaxation function in terms of a hypergeometric function,

_ I .R. 2 2 co(z) - - F(l/2, 1, ... +3/2, -O'\Iz ) .

z

(9.26)

The ~-terminator is an instrument of fine-tuning for the reconstruction of spectral densities with compact support and endpoint singularities that are known at least approximately. A minor problem is that the model relaxation function (9.26) is usually not found in computer libraries for ready use in applications of the ~­ terminator. The simplest means to generate it numerically on a line of points z = 3 £ - ioo turns out to be the continued fraction with coefficients (9.25). Convergence for 0 at T=O of the 1D 5=1/2 Heisenberg ferromagnet (6.22) with .1=1. The inset shows the corresponding sequence a k versus 1/k [from Viswanath and Muller 1991].

For our second example we consider the T=O spin autocorrelation function <S1(t)Sj> of the ID s=1I2 XX model (7.33). The associated spectral density «I»5Z(co) is derived from the exactly known dynamic structure factor, (6.38), by a sum over all wave numbers and can be evaluated in terms of elliptic integrals [Muller and Shrock 1984]. The spectral weight is confined to frequencies lcol < 2J and

124

Chapter 9

singularities are present at roll = 0,±1,±2. The Ak-sequence produced by the recursion method up to 1 -1) are allowed by exact sum rules. The main plot of Fig. 9-8 shows ak versus Ilk. That sequence is clearly not consistent with a pennissible power-law singularity. What we see in the plot is the signature of a &-function present at ro=O in an otherwise well-behaved spectral density with compact support. k

!1k

M(k)

k

!1k

M(k)

1

2

2

2

4

9

3 5

20

7

61

5

32 49

98

6

66 49

243

7

70 99

340

8

128 99

707

9

3 4

944

10

5 4

1775

11

112 143

2244

12

174 143

3891

3" 3

3" 4

'5

Table 9-1: Exact values of the first 12 coefficients Ak pertaining to the spin autocorrelation function at T=oo of the classical XXX dimer (6.52) with Jj=!t=Jt=1. The inset shows the same type of plot for the corresponding exLc)·sequence with «1>0(00) = 1/2.

M

o --

M(C) -

0

-

1i..

1

, (c)

M 2k = [l-qlo(oo)]M2k '

(9.41) k=I,2, ...

The relation between the two sets of continued-fraction coefficients is not that simple, but it can be shown that the sequence 1 A k = 2(~-1 +~), k=I,2, ...

(9.42)

generated from the moments M 2k of (9.40) does not depend on the value of the constant $0(00). Hence we have A k = A~C) for all k. For our current application, the !:ik from Table 9-1 yield A k=1 independent of k. However, it is the !:i~ctsequence that we wish to know, because it is from that s~uence that we can most likely extract quantitative information on the function $oc)(ro). For a given !:ik-sequence, we can determine the sequence !:i~c) pertaining to the spectral density $6C)(ro) from (9.41) and the recurrence relations (3.33-34), but only if we know the spectral

Section 9-9

127

weight of the o-function central peak, Le. the constant 4>0(00). That infonnation, however, is usually not available. In our current application, we can proceed as follows. We determine a whole set of ~£C)-sequences from the given ~k-sequence, where each member corresponds to a different value of the unknown constant 4>0(00). From that set of ~£c)-sequenceswe generate in turn a set of cx£C)-sequences according to (9.37). None of these sequences will converge to a finite value in a plot of a£c) versus 1Ik except the one for which the o-function central peak was given the correct amount of spectral weight. For the case at hand, we have 4>0(00) = 112. The corresponding a£C)_ sequence is plotted in the inset to Fig. 9-8. It tends to converge to the value a=1, implying that 4>~C)(ro) has a linear cusp singularity at 00=0. In fact the ~£c)-sequence resulting from 4>0(00) = 112 can be written in closed fonn as follows:

~(C)

= 1+_1_,

2k-1

2k+ 1

~ = 1 __1_. 2k+ 1

(9.43)

We recognize this to represent the special case 000=2, a=~=1 of the model ~k­ sequence (5.13). It is then a simple matter to reconstruct the exact autocorrelation function Co(t) = 3<sj(t)Sf> and its spectral density: 3 -4t 2

+ _ _cos(2t) , 4

(9.44)

4t

4>0(00) = 1to(ro) + 2:lrol(4-ot)EX2-lrol) . 8

(9.45)

9-9 Terminator with Matching Infrared Singularity After having established reliable means to detect the presence of infrared singularities in spectral densities with compact support and to detennine their nature, we should like to use tenninators for their reconstruction which are based on model spectral densities with matching singularities at 00=0. The result is a considerable improvement in the quality of the reconstructed spectral density. For a demonstation of this point, we return to the ~k-sequence shown in Fig. 9-7, pertaining to the T=O spin autocorrelation function <S](t)Sj> of the 10 s=1I2 XX model (7.33). Our analysis of that ~k-sequence in Sec. 9-7 has yielded two pieces of implicit infonnation that can be legitimately incorporated into the tennination function: (i) the bandwidth 1001 ~ 000 = 2J and (ii) the exponent value a=1 of the infrared singularity. A convenient ad hoc model spectral density with these properties is the continuum part of the result (9.45): 4>0(00)

1t 2 = _lrol(C%-ot)EXC%-lrol)

%

.

(9.46)

The associated model ~k-sequence and model relaxation function are both known in closed fonn:

128

Chapter 9

1.5

3'

1.0

'-"

N NO

>&

0.5

0.0 -iC---,....----..----r---,....----,-----r----r----j-"--' 2.0 1.0 0.0

Figure 9-9: Spectral density ~~Z(o» at T=O for the 1D s=1/2 XX model (7.33) with .1=1. The full line represents the result derived from the continued fraction (9.3) (with Z = E-iro, £=0.001) tenninated at level K=15 as described in the text. The dashed line represents the exact result (11.43) [from Viswanath and Muller 1991).

(9.47)

2) ( %( %J % We thus take the model relaxation function (9.48) with parameter value -

2z

co(z) ::: -

z2 % 1+n 1+-

2z

- -

•

(9.48)

Z2

000 = 21, detenmne the termination function f 15(Z) iteratively via (9.5), insert it into expression (9.3) along with the known coefficients A1,oo.,A 15 from Fig. 9-7, and evaluate the resulting spectral density via (9.7). The result is shown in Fig. 9-9 along with the known exact result. The agreement between the two curves is not perfect but very satisfactory if one takes into account that the reconstruction is based on a mere 15 numbers. The match is best at small 00, where both the exact spectral density and the model spectral density have the same singularity exponent, a=1, previously inferred from the Ak-sequence directly. The discrepancy is somewhat larger near ro=21, where the exact spectral density has a discontinuity, whereas the model spectral density goes to zero linearly. Despite this mismatch in

Section 9-10

129

singularity exponent, the reconstructed spectral density reproduces the discontinuity fairly well. The agreement between the two curves is worst near ro=J, where the exact result has one more singularity, but the model spectral density does not. The importance of the matching infrared singularity in the terminator used for this application is best illustrated by comparing the result shown in Fig. 9-9 with that previously obtained from a square-root terminator (see Fig. 8-6).

9-10 Compact a-Terminator When confronted with the task of reconstructing a spectral density whose IJ.k sequence tends to converge on an alternating path toward a unique and finite asymptotic value, we best proceed as follows: At first we estimate the asymptotic value IJ... directly from the IJ.k-sequence and the value of the singularity exponent a from the corresponding ak-sequence (9.37). Then we construct a termination function from a model spectral density with matching bandwidth and matching infrared singularity. For the application discussed in Sec. 9-9 we had a matching terminator conveniently at hand. For arbitrary bandwidths and singularity exponents we introduce here a new terminator, the compact a-terminator. Its model sr:;tral density is a special case of (5.12) and depends on the two parameters OOO=2IJ..!2 and a: (1)0(00)

= ~(1+a)loYO>olafXO>o-IOOI) 0>0

(9.49)

.

The model ~k-sequence is a special case of (5.13),

-

IJ. k

=

(2k-1 +a)(2k+1 +a) ot(k+ai (2k -1 +a)(2k + 1 +a)

(even k)

(9.50)

(odd k)

and the model relaxation function can be expressed in terms of hypergeometric functions: CO(z)

=

;z[F(l+a,1;2+a;iO\lZ) + F(1+a,1;2+a;-iO\lz)] .

(9.51)

If we set a=O, this terminator reduces to the familiar rectangular terminator (9.1719). There are other cases for which (9.51) can be expressed in terms of more elementary functions, for example,

co(z)

=

~[~ Tc%

+~J

I!ln(l 4z 1 -JiO\lz

+

+~J]

I;ln(l C% 1 -JO\Iiz

~

(a =-2.) 2

,

(9.52)

130

Chapter 9 5.0......------~-------------------,

1.4

0.0

(a)

-0.1

4.0

~

tS

1.2

-0.2

~

of the ID s=1/2 XX model (7.33), but now at infinite temperature. The first 20 tJ..k's as produced the recursion method in the Liouvillian representation (Sec. 6-4) are displayed in Fig. 9-IO(a). Their alternating approach with increasing k toward the limiting value t1.,,=J suggests the presence of an infrared singularity. Since the ~ converge from above and the tJ..2k_1 from below (opposite to the behavior observed in Fig. 9-6 for the corresponding T=O result), that singularity is expected to be divergent. A quantitative analysis of the associated

Section 9-11

131

ak-sequence (9.37) confinns this as shown in Fig. 9-1O(b). The ak's tend to converge toward a value somewhere between a.--o and a.=0.12. The exact result has, in fact, a logarithmic divergence [Katsura, Horiguchi and Suzuki 1970]. However, for our test application we are not supposed to use "insider knowledge". Therefore, we employ the compact a-terminator with our best estimates, 000 2, a -0.1, for the parameter values as inferred from Figs. 9-1O(a) and 9-1O(b). This implicit infonnation together with the explicit infonnation contained in the 35 known !1k's yields the spectral density shown as solid line in Fig. 9-1O(c). It agrees very accurately with the exact result (dashed line).

=

=

9·11 Gaussian Terminator In applications of the recursion method to interacting quantum many-body systems, it is frequently observed that the !1k-sequence of a specific spectral density under investigation has a linear growth rate, A.=1, as defined in Sec. 5-4. For the reconstruction of this type of spectral density from a finite number of known !1k's, we must choose a terminator which is consistent with that very property. The simplest model spectral density which does the trick is a Gaussian, d.. ((0)~ ""'0

=_ I4it _ e -cJ/cJo .

(9.54)

Cl\)

The associated model relaxation function is

-

co(z)

(ieZ2/cJo erfc(z/Cl\», =_

(9.55)

Cl\)

and the model Ak-sequence is strictly linear in k, 1 2 !1 k = -c4} . 2

(9.56)

with the adjustable parameter 000 determining the slope. Before we proceed with an application of the Gaussian tenninator, we should like to emphasize that in our method the ~h-Ievel tennination function f ~z) (at z = -ioo) is not itself a Gaussian except for K=O. In fact, it has a very strong Kdependence. This is dictated by the requirement that the tenninator simulates the correct average growth rate of the exact !1k-sequence. This important criterion is not taken into account if the termination function itself is modeled after a pure Gaussian as is the case, for example, in the Gaussian-memory-function approximation discussed in Sec. 8-5. For a first application of the Gaussian tenninator, consider the spin autocorrelation function of the 10 s=1I2 XX model (7.33) at T=O. This function has been analyzed on a rigorous basis (see Chapter 11 for a more detailed discussion). The spectral density, «1»0(00), as detennined on the basis of that rigorous analysis [Muller and Shrock 1984] is plotted as a dashed line in Fig. 9-11. It has

132

Chapter 9

three singularities on the frequency range shown: a square-root divergence at ro=O, a logarithmic divergence at OF=J, and a square-root cusp at «r=2J. The coefficients ~1""'~13' shown in the inset to Fig. 9-11 are clearly consistent with a linear growth rate (A.=l). The slope of the regression line detenriines the value of the parameter, CJ.lo=1.071J, of the Gaussian terminator. These ingredients are then put into our standard procedure for the reconstruction of the spectral density. The result is represented by the solid line in the main plot of Fig. 9-11. The outcome is very encouraging. The reconstructed spectral density reproduces all major features of the exact function at least qualitatively. Even more important is that our method has not produced any artificial features which may invite misinterpretation. The successful reconstruction of the spectral density CIlQ(ro) by means of the recursion method is an important test not only for the Gaussian terminator but for the recursion method itself, given the fact that this function exhibits many properties that are believed to be generic for spectral densities in quantum manybody dynamics [MUller and Shrock 1984, MUller 1987].

o. 0 -l-,....-,-.....,..............---r--r-'"'~.....,......'T""""""T---r-..,.-"........,......'T""""""T--r,.=;:=:;::=r==l 0.0

0.5

1.0

1.5

2.0

2.5

Figure 9-11: Spectral density CIl~X((O) for the spin autocorrelation function

- e -tit .

(10.1)

st,

If the interaction Hamiltonian conserves the total spin component S¥ = Ll we expect the same correlation function to exhibit a long-time tail, characterized by an algebraic decay law of the form <sr(t)Sr> - t-dJ2 ,

(10.2)

where d is the dimensionality of the lattice. In this situation, the dominant transport mechanism of spin fluctuations would be spin diffusion. The assumption is that for sufficiently long wavelengths and times, the time evolution of the dynamical variable SJ!(q,t)

=L

eiq""Sr(t)

(10.3)

I

is governed by a diffusion equation, (1004)

It must be said, however, that there exists as yet no rigorous theoretical derivation of this slow longitudinal spin motion from the rapid transverse spin motion, specified by the microscopic equations of motion for individual (classical or quantum) spins Si [Forster 1975, Fogedby and Young 1978]. The validity of the phenomenological equation (1004) implies that the correlation function of the variable (10.3) decays exponentially in time, <sJ!(q,t)SJ!(-q,O»

_ e-Dq2t ,

(10.5)

for small q and large t, and that the spin autocorrelation function decays algebraically as in (10.2). More details about the assumptions underlying expression (10.5) will be dicussed in Sec. 10-20 in the context of the classical Heisenberg model. In ID and 20 systems, this long-time tail, if indeed present, is in turn responsible for infrared divergences of the type

~J!(oo) _ (

00-

1/2

In(1/oo)

(10) (20)

(10.6)

in the corresponding spectral density. This quantity is directly amenable to experimental investigation, e.g. via NMR spin-lattice relaxation rates. Some experimental results will be discussed in Sec. 10-18.

I The prototype of this process is described, in the context of the recursion method, by the (n=1)-pole approximation of any dynamical system (see Chapter 8).

138

Chapter 10

10~2 10 s=112 XYZ Model on Semi-Immite Chain h is ironic but scarcely surprising that none of the exactly known functions <Slf(t)S'f> for quantum spin models is consistent with the expectations derived from the spin diffusion phenomenology. For classical spin systems too, most of the available simulation data exhibit significant deviations from predictions based on hydrodynamic assumptions (Secs. 10-14 to 10-23). Exact results for dynamic correlation functions of interacting quantum spin systems are scarce. Some such results have been reported in Chapter 7 for models with infinite-range interaction. With almost no exceptions, all other exact results pertain to the ID s=112 XY model. It is the special case lz=O of the ID s=1/2 XYZ model. The XYZ Hamiltonian for a semi-infinite chain reads

H XYZ

= -E {J?tS/:! /=0

+

lyS/S/:! + lzS/S/~d .

(10.7)

In the following the focus will be on the study of spin autocorrelation functions <Sl(t)S{>, Jl=X,y,z, at T=oo for this model. We shall review existing results and compare them with new results for bulk spins (/=00) and spins [=0,1,2,... at or near the boundary of the semi-infinite chain. Much attention will be given to the special case with uniaxial symmetry (Jx=ly=J), the XXZ model H xxz =

-E {J(stS/:! /=0

+ S/S/:!) +

l?/S/~!}

(10.8)

For each term alone, the dynamics can be analyzed exactly: the XX model (Jz=O) is equivalent to a system of noninteracting lattice fermions (Sec. 6-3), and the X model (l=O) is as trivial as the quantum harmonic oscillator (Sec. 6-1). For other parameter values, the T=oo dynamics of the XXZ model is quite complicated, and various interesting transitions between different types of dynamical behavior can be studied. For that purpose, the recursion method turns out to be an invaluable calculational tool. Although this Chapter focuses more on ph'ysical phenomena than on calculational techniques, due attention will be given to every new facet of the recursion method as it is introduced along the way. 10-3 Spin~1/2 XX Model: Neither Spin Diffusion nor Exponential Relaxation Let us begin our exploration of high-temperature quantum spin dynamics with the exactly solvable semi-infinite XX model H xx

= -lE

{sts /:! + s/s/:!} .

(10.9)

/=0

We are interested in spin autocorrelation functions <Slf(t)S'[> at T=oo and the associated spectral densities

Section 10-3

1111

_

4>0 (ro)[ =

f

-

dte

11 11 irot <S[ (t)S[ >

-00

_

, Il"""'x,y,z.

139

(l0.1O)

<SiSi>

For the bulk spin (1=00) these functions have been determined exactly many years ago. The results for Il=z, expressible in terms of a Bessel function and a complete elliptic integral, respectively,2 (lO.lla)

4>~(ro)oo = .?:-K(b -cJ14J 2 )a:1-cJI4J 2) , reJ

(1O.11b)

were first derived by Niemeijer [1967] and by Katsura, Horiguchi and Suzuki [1970]. In the fermion representation of the XX model, the evaluation of these quantities is straightforward e.g. in terms of a two-particle Green's function for noninteracting lattice fermions. Such a calculation was carried out in Sec. 6-3. Note that (lO.lla) decays more rapidly, _(1, than the diffusive long-time tail (10.2) with d= 1 does. Correspondingly, expression (l 0.11b) has only a logarithmic infrared divergence as opposed to the characteristic ro- 1I2-divergence of ID spin diffusion. The fluctuations of S;, in this model are obviously not governed by a diffusive process despite the conservation law Sf =const. The fluctuations of S/ also decay algebraically, _(112 with oscillations, rather than exponentially, -exp(-Dq2t), as would be expected for a diffusive process. A physical interpretation of this peculiar transport of spin fluctuations will be given in Sec. 10-11. The determination of the function <S;,(t)S;,> for that same model is far less straightforward. The exact result was first conjectured on the basis of a moment analysis for finite chains [Sur, Jasnow and Lowe 1975]. Rigorous derivations, based on the analysis of infinite Toplitz determinants, were reported within one year by Brandt and Jacoby [1976] and independently by Cape1 and Perk [1977]. The result is a pure Gaussian:

<S":(t)S":> = <S!,(t)S!,>

(l0.12a)

4>U«1» = o 00

(l0.12b)

2{it e -ofIJ2. J

It is remarkable that this result can be derived with little effort from the recursion

method in the spin representation [Florencio and Lee 1987] as reported in Sec. 6-4.

brhroughout this Chapter, the frequency moments of (10.10) as defined in (3.30) and the associated continued-fraction coefficients will be labelled accordingly, ~~(~ and !i'tt(~, respectively.

140

Chapter 10

The Gaussian decay of (l0.12a) is anomalous again. A nonnal relaxation process would be characterized by exponential decay at long times. The non-generic processes that govern the transport of spin fluctuations in this model are further indicated by the fact that all pair correlations is readily recognizable by the bounded support of the spectral density (l0.11b). That same conclusion cannot be drawn from a mere inspection of the results (10.12). Spectral densities with unbounded support are typical for the dynamics of interacting degrees of freedom. In order to detect the free-particle nature of the XX model in the xx-autocorrelation function, it is necessary to study boundary effects.

10-4 Boundary Effects: Buildup of an Infrared Divergence The zz-autocorrelation function was determined in closed fonn for all sites on the semi-infinite chain [Gon~alves and Cruz 1980; Brandt and Stolze 1986]: <S/(t)S/> ::: 2.[Jo(Jt) -( _1)l+1 JZ(l+I)(Jt)]z .

4

(10.13)

In the bulk limit l~oo, only the first tenn in the square bracket survives, and the result (lO.lla) is recovered. The Fourier transfonn of the Bessel function In(Jt) is nonzero only on the interval [-J, 1]. The spectral density ~5Z(CO)l associated with (l0.13) is thus confined to the interval [-21,21]. The long-time asymptotic expansion (LTAE) of the function (10.13) was found to have the following general structure [Stolze, Viswanath, and Muller 1992]: <S/(t)S/> -

E- a;t-(2n+3) n=O

-

+ (eziJtE b;(it)-(n+3) + c.c}.

(l0.14)

n=O

The leading terms determine the dominant singularities ip the spectral density ~5Z(CO)l: quadratic cusp singularities at the endpoints (co = ±2J) and at ro=O, for all 1. The explicit result for 1=0, expressed in terms of complete elliptic integrals reads [Stolze, Viswanath, and Muller 1992]:

~~(co)o:::

128(1 +oY21)f(l +cJ/4J Z)E(21-CO) - CO K ( 21-CO]. (l0.15) 21+co J &1:21+CO

31CJ

L

The shape of the spectral density ~5Z(CO)l for selected values of 1 is shown in Fig. 10-1. The bulk limit is subtle: the quadratic cusps at the endpoints become steeper and steeper and, for l~oo, transfonn into the discontinuities displayed by (lO.llb). Likewise, the maximum at ro=O grows higher and narrower and, for l~oo, turns into a logarithmic divergence. In Chapter 9 we have already predicted some of these spectral properties in test applications of the recursion method. The quadratic cusp at the band edge of (l0.15) was determined in Sec. 9-4 from an extrapolation of the ~k-sequence (9.23) (see Fig. 9-3). The spectral density ~5Z(co)o itself was then reconstructed in Sec. 9-5 from the first 50 known continued-fraction coefficients Llf(O) by means of the ~-

Section 10-4

141

terminator (see Fig. 9-4). For spins near the boundary of the semi-infinite chain (~l), the recursion method yields ~k-sequences that tend to converge to the same value ~;,z(l) = J2, but their approach is alternating in character initially and then crosses over to uniform convergence. This is illustrated for the case 1=1 in Fig. 10-2 (inset). The associated ~k-sequence still converges to the same value ~=2 but much less uniformly than for the case of the boundary spin (1=0). Hence the ~-terminator (with ~=2 and 000=2) is still applicable for the reconstruction of the spectral density $ijZ(oo)l. The result of that calculation is shown in Fig. 10-2 (solid line) together with the exact result (dashed line) taken from Fig. 10-1. The two curves are almost indistinguishable. The emerging alternating character of the ~k-sequence for 1 ~ 1 reflects the buildup of the logarithmic divergence at ro=O in the bulk limit (l~oo). In Sec. 9-10 we have already analyzed that singularity by extrapolating the a ksequence (Fig. 9-1O(b)). Furthermore, we have reconstructed the spectral density (I0.11b) from the known ~k's by using the compact a-terminator (Fig. 9-IO(c)). 7.0-.-------------------------.

6.0

5.0

2.0 1.0 0.0 -¥--O:::::::""-r-----,----.,.----,...---.----r----.-----=::::~ 2.0 1.5 0.5 1.0 -2.0 -1.5 -1.0 -0.5 0.0

Figure 10-1: Spectral density $~Z(co), at T=oo of the semi-infinite 10 8=1/2 XX model (10.9) with ..1=1 as determined by the Fourier transform of expression (10.13). The four curves represent the cases /=0 (boundary spin), /=1,5 and /=00 (bulk spin) [from Stolze, Viswanath and MOller 1992].

142

Chapter 10

5.0...------------------------, 1.5-.------------,

4.0

-

...... '-'

~~

1.0

~X(co)p 1=1,2,3,4, at T=oo of the semi-infinite 1D 8=1/2 XX model (10.9) with .1=1. The maximum value of kis 55,28,22,20, for the four cases, respectively. The solid line represents the sequence 6.f(oo) =k/2 for the bulk spin case. The arrows indicate the limiting values 1/6.~X(~ for 1=1,2,3,4 [from Stolze, Viswanath, and Muller 1992].

10·6 Spin·112 XXZ Model Consider first the bulk-spin autocorrelation function <S;'(t)S;'> at T=oo of the 10 s=1I2 XXZ model (l0.8). The nontrivial but exactly solvable case lz=O (XX model) is an ideal starting point for the analysis of the cases lz:t:O by means of a general method which can produce the exact result in the XX limit. The recursion method is such a method as we have demonstrated in Sec. 6-4. Its strength for the analysis

146

Chapter 10

of the function <S;'(t)S;'> derives from the fact that gradual deviations from the exactly solvable limit lz=O produce only gradual deviations from the linear sequence,

Ll~(oo)

=

(10.21)

!.-12k , 2

associated with the Gaussian spectral density (10.l2b). These gradual changes in the Llk-sequence in turn produce gradual changes in the spectral density c1>o(ro)oo' which can be analyzed quantitatively.

5.5 -r-----;:========================::::;--------i 1.2

4.5

.
for Jz=O is non-generic in that respect, attributable to the free-fermion nature of the XX model. The expectation is that a slight increase of Jz from zero causes a crossover in <S;,(t)S;'> from Gaussian decay to exponential decay. It would be impossible to detect such an effect directly in the ~k-sequence, because a transition from Gaussian decay to exponential decay at long times does not by itself have much impact on the shape of the spectral density. However, given the simplicity of the exact result (10.12a), the crossover might be discernible in a short-time expansion of <s;'(t)S;'>. This is indeed the case [B6hm et al. 1994]. In Fig. 10-6 we have plotted upper and lower bounds of the function In(<S;,(t)S;'»/ (Jt<S;'S;'» versus Jt for four different parameter values of the XXZ model near the XX limit. The dashed line with slope -1 represents the pure Gaussian (10. 12a). Pure exponential decay would be represented by a horizontal line with negative intercept. The results for Jz#J show strong indications that the decay is slower than Gaussian and consistent with exponential decay (convergence toward a negative constant). Power-law decay would imply convergence toward zero. Whether or not the observed exponential decay represents the true asymptotic behavior is, of course, beyond the reach of this type of analysis. 10-8 Analysis of ~k·Sequences with Growth Rates near A.=1 In Secs. 9-11 to 9-13 we have already discussed ways to analyze ~k-sequences with growth rate A.=1 for the double purpose of (i) reconstructing spectral densities with the Gaussian terminator or the unbounded a-terminator and (ii) estimating infrared exponents on the basis of the explicitly known functional dependence (5.26) of the model coefficients ~k on the singularity of the model spectral density (5.24).

148

Chapter 10

-1.6..,....;:-------------------------,

-1.8

-2.4 J /J=O

z

-2.6 1.6

1.8

2.0

2.2

2.4

2.6

Jt Figure 1~: Short-time expansion of the bulk-spin autocorrelation function Co(~ =

<S~(~~>/<S~S~> at T=oo of the 10 &=1/2 XXZ model (10.8) for three parameter values

JJJ = 0.02, 0.05, 0.1 (solid lines) near the exactly solvable XX case (dashed line). The data are plotted in a way suitable for visualizing the crossover from Gaussian decay (negative unit slope at zero intercept) to exponential decay (zero slope at negative intercept). Each result of the short-time expansion Is represented by two curves corresponding to an upper and a lower bound of the function. The bounds have been determined from 14 exact frequency moments ~oo) [from Bohm et al. 1994].

In applications to ak-sequences with M:1, as they occur in the situation now under consideration (see Fig. 10-5), such an analysis should be carried out on the basis of the more general model spectral density (5.27). This remains impractical as long as we lack closed-form expressions for the continued-fraction coefficients ak pe.rtaining to (5.27) as functions of the three parameters Olo, <X, A.. However for growth rates sufficiently close to 1..=1, we can approximate the (M:1)-problem with a (I..=l)-problem if we replace the ak-sequence by the rescaled sequence A* _

A

ilk - ilk

1/A.

(10.22)

and then proceed as outlined in Secs. 9-11 to 9-13. The main distortions in the reconstructed spectral density caused by this approximation are of two kinds: (i) a change in the large-ro decay law and (ii) a change in the frequency scale. Whereas the former has only a negligible impact on the shape of the spectral-weight

Section 10-9

149

distribution, the latter may warrant attention and lead to significant improvement upon proper adjustment. 10-9 From Exponential Relaxation to Diffusive Long-Time Tails Let us now present the results for the bulk-spin spectral density :(00)_ at T=- of the 10 8=1/2 XXZ model (10.8) with .1=1 as reconstructed from the continued-fraction coefficients 4f("::').....4f:(oo) and a Gaussian terminator. The calculation was carried out by the use of the 4 k-sequence in the role of the original 4 k-sequence and then as outlined in Sec. 9-11. The six curves plotted at 0>0 to values 0.6,.... 1.0 [from Bohm et al. 1994).

150

Chapter 10

The further development of the spectral density as JIJ approaches the XXX case is shown by the five curves on the right. The shoulder becomes more pronounced, and the strong peak at 00=0 signals the presence of an infrared divergence for Jz=J, in accordance with spin diffusion phenomenology. The curve for the XXX case is in qualitative agreement with previous results obtained from finite-chain calculations [Carboni and Richards 1969, Groen et al. 1980], and by a calculation which uses the first two frequency moments of the dynamic structure factor in conjunction with a two-parameter diffusivity [Tahir-Kheli and McFadden 1969]. Note that the infrared singularity in ~(ro).o which is strongly suggested by the curves for JIJ'" 1 in Fig. 10-7, is in no way artificially built into our approach. It is a structure resulting solely from the 14 known continued-fraction coefficients. The results shown in Fig. 10-7 are expected to be most accurate for small values of JIJ, where the growth rate is closest to A.=l (see Fig. 10-5, inset). As the growth rate increases toward 1.... 1.22, the curves are likely to become subject to some systematic errors as explained in Sec. 10-8. We have estimated the systematic error in frequency scale not to exceed 2% for the curves at ogIJ~0.5 and 12% for those at O.5gIJ~l. Once we have recognized the strong indications for the presence of an infrared singularity, we can investigate its nature more quantitatively by a direct analysis of the explicitly known continued-fraction coefficients. In Sec. 9-12 we have outlined and tested such a method for ~k-sequences with growth rate A.=l. In the present context, the analysis must be carried out for the associated ~;-sequence as defined in (10.22). The results of such an analysis are compiled in Fig. 10-8. The circles represent the mean exponent a as a function of JIJ ranging from the XX model (Jz=O) to the XXX model (Jz=J) and somewhat beyond. The error bars indicate the statistical uncertainty for each data point, which is due to the fact that the analysis is based on a finite number of known continued-fraction coefficients. On top of the statistical error, the data are likely to be subject to a systematic error whose potential impact increases with the deviation of the .growth rate from A.= 1. As Jz approaches zero, both types of uncertainties (statistical and systematic) become smaller and disappear. The data point a(O)=O is exact and describes the spectral density (lO.l2b), which has no infrared singularity. The dependence on Jp ofthe mean exponent values displayed in Fig. 10-8 is quite remarkable in spite of the limited overall accuracy. The data strongly indicate that the function a(JIJ) stays zero over some range of the anisotropy parameter. A vanishing exponent at small but nonzero JIJ is consistent with and thus reinforces the conclusion reached from the short-time analysis in Sec. 10-7 that the function <S;'(t)S;'> decays faster than a power-law, namely exponentially. While the data point at JIJ = 0.5 is still consistent with a=0, the mean avalues have already a strongly decreasing trend at this point. A minimum value is reached exactly at the symmetry point (Jz=J) of the XXX model - the only point for which the conservation law Sf LiSI const holds, and therefore the only point for with one expects a diffusive long-time tail in <S;'(t)S;'>. Upon further increase of JIJ, the data points rise again toward a=O as expected. The minimum exponent value, a = -0.37±0.12, obtained for the XXX case is only marginally consistent

=

=

Section 10-10

151

with the standard value, cx=-1I2, predicted by spin diffusion phenomenology. That discrepancy is likely attributable to the systematic error in our data. 0.1

0.0

-0.1

~

-0.2

-0.3

-0.4

-0.5 0.0

0.5

1.0

J

z

Figure 10-8: Infrared-singularity exponent <X vs anisotropy parameter Jz of the spectral density CI>~X«(J)_ at T=oo of the 10 5=1/2 XXZ (10.8) with ..1=1. The data points were obtained from the coefficients af(oo),...,af:(oo) by analyzing the a~-sequence as outlined in Sec. 9-12 [from B6hm et al. 1994].

10-10 Sustained Power-Law Decay The conservation law Sf LjS7 const for the spin fluctuations in z-direction holds over the entire parameter regime of the XXZ model (10.8). Consequently, the longtime behavior of the correlation function <S;,(t)S;,> or the low-frequency behavior of the spectral density ijZ( 00)00 is expected to be much less affected by the symmetry change of Hxxz at lz=1 than the functions <S;,(t)S;,> and 0(00)00 were. The verification of sustained power law decay in <S;,(t)S;,> at lz# as a contrast to the result presented in Secs. 10-7 and 10-9 will further support the case for quantum spin diffusion. Here we encounter a problem which prevents us from carrying through the calculation for parameter values near the exactly solvable XX limit (1/1=0). The breakdown is caused by a crossover in the growth rate of the relevant /1k-sequence. Figure 10-9 shows the sequences of /1iZ( 00) for four different parameter values.

=

=

152

Chapter 10

Between JjJ=o.6 and JjJ=1.0, the sequence of known coefficients has a welldefined growth rate somewhat higher than A.=1. For the XX model (Jz=O), on the other hand, growth rate A.=O is well known to be realized (see Sec. 10-4). The sequence for JjJ=O.1 has attributes of both regimes. It starts out with A.=o up to k-7 and then begins to grow with A.?1, thus causing a kink in tik vs k. It is impossible to analyze such sequences on the basis of a unique value of A, and, therefore, impossible to carry out the analysis described before without major modifications. 3

10.0

B.O

-8

.......... N

6.0

N~

at T:oo of the 1D 5=1/2 XXZ model with Jz = 0 (XX case), J z = 0.1,0.6, and J z = 1.0 (XXX case). The kink in the sequence for J z = 0.1 illustrates the crossover between growth rates A. 0 and A.